Curves on non--orientable surfaces and crosscap transpositions
S. Öykü Yurttaş




Curves  on non–orientable  surfaces and crosscap transpositions   ]
 Curves  on non–orientable  surfaces and crosscap transpositions    




Öykü Yurttaş]Öykü Yurttaş

Dicle University Science Faculty Mathematics Department
21280, Diyarbakır, Turkey

Primary 57N16; Secondary 57N05






Let  N_g,n be an n–punctured non–orientable surface of genus g with one boundary component. For g≥ 2 one of the generators of the mapping class group of N_g,n is a crosscap transposition.  We give explicit formulae for the action of crosscap transpositions and their inverses  on the set  of multicurves in N_g,n in terms of  generalized Dynnikov coordinates.     

 


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    December 30, 2023
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§ INTRODUCTION

 


		


 Let N_g,n  (g≥ 2) be  a non–orientable surface of genus g with n punctures and one boundary component.    In all figures of this paper  each disk with a cross represents a crosscap, a graphical representation of a Möbius band. This means that  the interior of each such disc is removed, and the antipodal points on the resulting boundary component are identified. Throughout, we take a standard model of  N_g,n where the punctures and the crosscaps are arranged  along the horizontal diameter of N_g,n as shown in Figure <ref>.  A simple closed curve in N_g,n is  inessential if it bounds an unpunctured disk, once punctured disk or an unpunctured annulus.  It is essential, otherwise. If a regular neighborhood of an essential simple closed curve in  N_g,n is an annulus it is called  2-sided, and if it is a Möbius band it is called 1-sided.  We call  the core curves and  the double covers of  the core curves  Möbius curves.  A multicurve in N_g,n is a disjoint union of finitely many essential simple closed curves in N_g,n modulo isotopy.  We write 𝔏_g,n to denote the set of multicurves in N_g,n. 


Multicurves on orientable surfaces are usually described by techniques such as the Dehn–Thurston coordinate system <cit.>. An alternative way to describe multicurves on finitely punctured disks is to use the Dynnikov coordinate system <cit.>.  In 2016, Papadopoulos and Penner <cit.>  provided analogues for non–orientable surfaces of several results from Thurston theory of surfaces including the Dehn-Thurston coordinate function. Inspired by their work, the generalized Dynnikov coordinate system  was introduced in <cit.> for multicurves in N_g,n which provides an explicit bijection between 𝔏_g,n and a certain subset of (ℤ^2(n+g-2)×ℤ^ g)∖{0}.  Here, we give a modified version of the generalized Dynnikov coordinate system together with the formulae in Theorem <ref> (a corrected version of Theorem 2.14 in <cit.>) for the inverse of the Dynnikov coordinate function.  Furthermore, with a slight modification, we also describe generalized Dynnikov coordinates for multicurves in N_g,0, which wasn't covered in <cit.>. Let n >1. The generalized Dynnikov coordinates can be described as follows:

  

Let 𝒜_g,n be the set of arcs α_i (1≤ i≤ 2n-2), γ_i (1≤ i≤ 2g-2) and β_i (1≤ i≤ n+g-1) as depicted in Figure <ref>: the arcs α_2i-3 and α_2i-2  (2≤ i≤ n) join the i-th puncture to the boundary,  the teardrops γ_2i-1 and γ_2i encircle the i–th crosscap and have  endpoints on the boundary, and the arc β_i (1≤ i≤ n-1) has endpoints on the boundary and passes between the i–th and (i+1)–th punctures, β_n  passes between the n–th puncture and the first crosscap, and  β_n+i  (2≤ i≤ g-1) passes between the i–th and (i+1)–th crosscaps. Finally, c_i (1≤ i≤ g) denotes the core curve of the i–th crosscap. 




 

  Given ∈𝔏_g,n  let L be a minimal representative of  (that is, L intersects each of the arcs and curves minimally). For the sake of brevity, let α_i, β_i,  γ_i also denote  the number of  intersections of L with the  corresponding arcs.  We write c_i=-1 if  contains the i–th core curve, c_i=-2m if  contains m disjoint copies of the double cover of the i–th core curve and c=-2m-1 if  contains m disjoint copies of the double cover of the i–th core curve  plus the core curve itself.   Otherwise c_i denotes  the number of  intersections of L with the core curve of the i–th crosscap.  It will always be clear from the context  whether the symbols α_i, β_i,  γ_i  and  c_i  refer to arcs and curves rather than to integers. We write (α; β; γ,  c)∈ℤ^3n+3g-5∖{0} for the collection of  these integers associated with . Let x^+=max(x,0) throughout the text. 
   



 Let the function ρ𝔏_g,n→  (ℤ^2(n+g-2)×ℤ^ g)∖{0}  be defined by 

    ρ(ℒ) = (a;  b; t;  c):=(a_1, …,a_n-1; t_1,…, t_g-1;  b_1, …, b_n+g-2;  c_1,…, c_g)
 


where 

    a_i   =  
     α_2i-α_2i-1/2; 1≤ i≤ n-1, 
     t_i   = γ_2i-γ_2i-1/2; 1≤ i≤ g-1, 
    
    b_i   =β_i-β_i+1/2; 1≤ i≤ n+g-2


We say that (a;  b; t;  c)  are the generalized Dynnikov coordinates of ∈𝔏_g,n.
Let ψ_i=max(c_i^+-|b_n+i-1|, 0) (the use of this parameter will be explained later) and 𝒮_g,n={ (a;  b; t;  c)∈ (ℤ^2(n+g-2)×ℤ^g)∖{0}: |t_i|+ψ_i   is even for   1≤ i≤ g-1}.
 
 
 Note the special case n=1 where there is no a_i coordinate, and the special case g=1 where there is no t_i coordinate. 
 
 
 The intersection numbers  (α; β; γ,  c) (and hence the multicurve ) can be recovered from the generalized Dynnikov coordinates  (a;  b; t;  c)∈𝒮_g,n.  Theorem <ref> gives the inverse of the generalized Dynnikov coordinate function by presenting a formula that describes multicurves from given generalized Dynnikov coordinates.

 
 
 


  
 


 
	
Let (a;  b; t;  c)∈𝒮_g,n. Then (a;  b; t;  c) corresponds to a unique multicurve in ∈𝔏_g, n which has 


    α_i   = {[  (-1)^i a_⌈ i/2 ⌉+β_⌈ i/2⌉/2                             ; (-1)^i a_⌈ i/2⌉+β_1+⌈ i/2⌉/2                              ].
    γ_i        = {[ (-1)^i t_⌈ i/2 ⌉+β_n+⌈ i/2⌉-1+ψ_⌈ i/2 ⌉                                        ;    (-1)^i t_⌈ i/2⌉+β_n+⌈ i/2⌉+ψ_⌈ i/2 ⌉                                         ].
    β_i   = Z_i+2max(0,c_g-Z_n+g-1/2)

 





where

    X_i   =2[|a_i |+max(b_i,0)+∑^i-1_k=1 b_k]
    
    Y_i   =2[max(b_n+i-1, 0)+∑^n+i-2_k=1b_k]+|t_i |+ψ_i
    
    Z_i   =max_1≤ s ≤ n-1
    
                      1≤ t≤ g-1{X_s,Y_t}-2∑^i-1_k=1b_k

 Here ⌈ x ⌉ denotes the smallest integer which is not less than x.  
	
	
	
	
	 The main goals of this paper are first to prove Theorem <ref> (correcting the proof of Theorem 2.14 in <cit.>); and then give the derivation of the formulae in Theorem <ref> which describes  how generalized Dynnikov coordinates change under the action of crosscap transpositions u_i and u^-1_i (1≤ i≤ g-1). Therefore, Theorem <ref> computes for each mapping class β written as a word of crosscap transpositions, β:𝒮_g,n→𝒮_g,n given by,
β(a; t; b; c)=ρ∘β∘ρ^-1(a; t; b; c). A crosscap transposition u_i is a generator of the mapping class group of N_g,n (i.e. group of isotopy classes of homeomorphisms of N_g,n) exchanging crosscaps ⊗_i and ⊗_i+1 in the counterclockwise direction  keeping each of the remaining crosscaps fixed <cit.>. We note that while the formulae in Theorem <ref> and Theorem <ref> seem to have a complicated form,  the method we use to obtain them is transparent since it purely relies on algebraic calculations and the properties of multicurves in terms of their associated  intersection numbers (α; β; γ,  c). In addition,  the formulae are ideally suited for computer implementation. 

 	
	
 For computational and notational
convenience, we will work in the max-plus semiring (ℝ,max,+) equipped with the additional and multiplicative operations a+ b = max(a,b) and a× b = a+b  to obtain the formulae in Theorem <ref>.  As we  use the normal notation of addition, multiplication, and division  we enclose the formulae in square brackets to indicate that these will be interpreted in the max–plus sense. Therefore, [a+b] = max(a,b), [ab]= a+b, [a/b] = a-b, [1]=0. 



To prove Theorem <ref> we shall make use of particular arc systems  called clovers and scales, each of which is associated with an exceptional parameter, certain linear combinations of generalized Dynnikov coordinates, denoted d_i, e_i, e̅_i, f_i,f̅_i, g_i and g̅_i.  




For notational convenience we write B_j=2b_j (i.e. [B_j]=[b^2_j]) in Theorem <ref>.





Let ∈𝔏_g,n have generalized Dynnikov coordinates (a;  b; t;  c).  Let (a';  b'; t';  c') and  (a”;  b”; t”;  c”) be the generalized Dynnikov coordinates of u_i() and u^-1_i() respectively. Then a'_j=a”_j=a_j, b'_j=b”_j=b_j for all 1≤ j≤ n;  (c'_i, c'_i+1)=(c_i+1,c_i) and (c”_i, c”_i+1)=(c_i+1,c_i) for 1≤ i≤ g-1; and for 1≤ i≤ g-3 we have




    t_i'    =  g_i(t_i(1+d_iB_n+i-1)+d_iB_n+i-1t_i+1) 
         t_i+1'    = t_it_i+1B_n+i/f_i(t_i(d_i+B_n+i)+ d_it_i+1)
    t_i”   = [t_it_i+1/g̅_̅i̅(t_id_iB_n+i-1+t_i+1(d_iB_n+i-1+1))]
     t_i+1”   = [f̅_̅i̅(t_i+1(B_n+i+d_i)+d_it_i)/B_n+i]
            
       B_n+i'    = e_i/d_iB_n+i(t'_i+t'_i+1/t'_i+1)t_i/t_i+t_i+1
           B_n+i-1'    = B_n+iB_n+i-1/B'_n+i
    B”_n+i-1   =[B_n+iB_n+i-1/B”_n+i],
     B”_n+i   =[e̅_̅i̅/d_iB_n+i(t”_i+t”_i+1/t”_i)t_i+1/t_i+t_i+1]


In the special case the formulae above is interpreted as 


    t_g-1'    = g_g-1(t_g-1+d_g-1B_n+g-2(1+t_g-1))
    
    t_g-1”   = t_g-1/g̅_g-1(d_g-1B_n+g-2(1+t_g-1))      
      B_n+g-2'    = d_g-1/e_g-1B_n+g-21+t_g-1/t_g-1(1+t'_g-1)
    
    B_n+g-2”   = d_g-2/e̅_g-2B_n+g-2t”_g-1(1+t_g-1)/1+t”_g-1.

In all other cases t'_j=t_j, t”_j=t_j and b'_j=b”_j=b_j.  



The paper is organized as follows.  Section <ref> provides background material and contains a detailed study of  generalized Dynnikov coordinates giving proofs of Theorem <ref> and Theorem <ref>. In Section <ref>,  we introduce the notions of clovers and scales, certain collections of adjacent arcs in 𝒜_g,n and their images under u_i and u^-1_i  from which we obtain clover and scale  equalities  given in Lemma <ref>, Lemma <ref>, Lemma <ref>  and Lemma <ref>  which play key roles in the derivation of the formulae in Theorem <ref> which we also prove  in Section <ref>. 






§ CONSTRUCTING MULTICURVES FROM GENERALIZED DYNNIKOV COORDINATES

 In this section we prove Theorem <ref>  recalling basic properties that  a minimal representative  L  satisfies in terms of  the intersection numbers (α; β; γ,  c).  Let 1≤ i≤ n-1 and S_i denote the region bounded by  the arcs β_i and β_i+1 containing puncture i+1. We denote  by S_0 the left most region bounded by β_1 and the boundary  containing the first puncture.  Now, let 1≤ i ≤ g. Then S'_i denotes the region bounded by  the arcs β_n+i-1 and β_n+i containing the i–th crosscap.  S'_g denotes the right most region bounded by β_n+g-1 and the boundary containing the last crosscap.  Since L is minimal, there are finitely many connected components of L∩ S_i and L∩ S'_i which are depicted in Figure <ref> and Figure <ref>.
 
 
 
 


 
 
 









	
	
  * An above component of  L∩ S_i (1≤ i ≤ n-1)  has one endpoint on each of the arcs β_i and β_i+1, and intersects α_2i-1 but not α_2i. An above component  of  L∩ S'_i has one endpoint on each of the arcs β_n and β_n+1, and  intersects γ_2i-1 but not γ_2i.
	
	


  * An below component of  L∩ S_i (1≤ i ≤ n-1)  has one endpoint on each of the arcs β_i and β_i+1, and intersects α_2i-1 but not α_2i. An below component  of  L∩ S'_i  has one endpoint on each of the arcs β_n+i-1 and β_n+i, and  intersects γ_2i but not γ_2i-1.



 
  * A left loop component of L∩ S_i has both endpoints on β_i+1, and intersects both of  the arcs α_2i-1 and α_2i. A  left loop component of L∩ S'_i  has  both endpoints on β_n+i, and intersects both of  the arcs γ_2i-1 and γ_2i. If it intersects the core curve, it is called a left core loop. Otherwise it is called a left non-core loop.
 
 

 
 
     
  * A right loop component of L∩ S_i has both endpoints on β_i, and intersects both of the arcs α_2i-1 and α_2i.   A right loop component of L∩ S'_i  has both endpoints on β_n+i-1, and intersects both of  the arcs γ_2i-1 and γ_2i.  If it intersects the crosscap, it is called a right core loop of  L∩ S'_i. Otherwise it is called a right non-core loop of L∩ S'_i. 
     
      

      
    
  * A straight component of L∩ S'_i  has one endpoint on each of the arcs β_n+i-1 and β_n+i, and intersects the core curve and both of  the arcs γ_2i-1 and γ_2i.
      
      
      
    Above, below and loop components of  L∩ S_i and L∩ S'_i  are depicted red, green and blue respectively in Figure  <ref> and Figure  <ref>.   Straight components of L∩ S'_i are depicted purple in Figure <ref>(c) and Figure <ref>(d). Observe that there can only be left loop components in S_0 and right loop components in S'_g.  The following lemma gives two important equalities which are obvious from Figure  <ref> and Figure  <ref>. 
      
      
Let L be a minimal representative of a  multicurve ℒ∈𝔏_g,n  with intersection numbers (α; β; γ,  c).  Let ψ_i denote the number of straight components of  L∩ S'_i. Then, 

    max(β_i, β_i+1)   =α_2i-1+α_2i



    max(β_n+i-1, β_n+i)   =γ_2i-1+γ_2i/2-ψ_i




      
      
      
              
    
    
    	Given a minimal representative L of ∈𝔏_g,n we can initially observe that every component of L intersects each β_i and hence each α_2i-1∪α_2i and  γ_2i-1∪γ_2i an even number of times.  Therefore a_i, t_i and b_i are integers.



	
	For each  1≤ i ≤ n+g-2 let b_i=β_i-β_i+1/2. Then there are |b_i|  loop components in S_i ( 1≤ i ≤ n-1) and |b_n+i-1| loop components in  S'_i (1≤ i ≤ g-1). If b_i>0 the loop components are right and if b_i<0 the loop components are left. 
	
	
		


We prove the statement for S_i (the argument for S'_i is identical). Let 1≤ i ≤ n-1.  We first note  that there cannot be both left loop and right loop components in S_i  since the curves are mutually disjoint. Assume without loss of generality that β_i+1≥β_i. Observe from Figure <ref>(b)  that the additional intersections on β_i+1 come from left loop components in S_i since above and below components intersect both β_i and β_i+1 the same number of times. Since each left loop intersects β_i+1 twice it follows that there are  -b_i=β_i+1-β_i/2 left loop components in S_i. 









The number of loop components of L∩ S_0 is given by β_1/2, and the number of right loop components of L∩ S'_g is given by β_n+g-1/2.




 Let 1≤ i< n+g-2, and λ_i,  λ_c_i and ψ_i denote the number of non-core loop, core loop and straight components of L∩ S'_i. Then,


    ψ_i   =max(c_i-|b_n+i-1|,0)
    λ_i   =max(|b_n+i-1|-c_i, 0)   ,  λ_c_i= |b_n+i-1|-λ_i











There are three possibilities for a connected component of L∩ S'_i which intersects the crosscap. It can be a left core loop or a right core–loop  or  a straight core component.  Observe from Figure <ref> that we have 

    c_i   =ψ_i+ λ_c_i


    |b_n+i-1|   =λ_i+λ_c_i.


 If c_i-|b_n+i-1|≥ 0,  there exist components of L∩ S'_i other than core loop  components which intersect the crosscap. Such components can only be straight components and hence ψ_i>0 and λ_i=0 since non–core loops and straight components cannot exist at the same time. Then, |b_n+i-1|=λ_c_i  and hence  ψ_i=c_i-|b_n+i-1|  by  (<ref>) and (<ref>).  If c_i-|b_n+i-1|<0, then there exist  non–core loop components as well as the core loop components. That is,  λ_i>0 and hence ψ_i=0. Therefore,  c_i=λ_c_i and hence λ_i=|b_n+i-1|-c_i by  (<ref>) and (<ref>).  Therefore, we get 
 ψ_i=max(c_i-|b_n+i-1|,0) and λ_i=max(|b_n+i-1|-c_i, 0) as required. We immediately get from  (<ref>) that     λ_c_i= |b_n+i-1|-λ_i. 
 

  Let 1≤ i< n+g-2. There are b^+_i=max(b_i, 0) right loops  and (-b_i)^+=max(-b_i, 0) left loops  about crosscap i.  By Remark <ref>  there are only right loop components of  L∩ S'_g and the number of those is given by β_n+1/2. It immediately follows that there are c_g core loops and hence λ_g=β_n+g-1/2-c_g  non–core loops of L∩ S'_g.
 
  



The following Lemma is obvious since each above and below component in S_i intersects α_2i-1 and α_2i, and  each above and below component in S'_i intersects γ_2i-1 and γ_2i respectively (see Figure <ref> and Figure <ref>).


Let there be A_i and B_i (1≤ i ≤ n-1) above  and  below components of L∩ S_i; and A'_i and B'_i (1≤ i ≤ g-1) above and  below components of L∩ S'_i  respectively. Then,  

    A_i   =α_2i-1-|b_i|
    
    B_i   =α_2i-|b_i|
    
    A'_i   =γ_2i-1/2-ψ_i-|b_n+i-1|
    
    B'_i   =γ_2i/2-ψ_i-|b_n+i-1|




 The curve in Figure <ref> has A'_1=A'_2= B'_1=0; B'_2=2; λ_1=λ_2=0, λ_3=1; λ_c_1=λ_c_2=1=λ_c_3=1; 
 ψ_1=ψ_2=0. These  parameters will frequently be referred to  throughout the paper. 

The generalized Dynnikov coordinate function ρ:𝔏_g,n→𝒮_g,n is a bijection:   To describe its inverse, it is sufficient to describe a function from 𝒮_g,n to ℤ^3n+3g-5∖{0}. It is easy to check that this function sends each (a;  b; t;  c)∈𝒮_g,n to the intersection numbers (α; β; γ,  c) associated with a multicurve  with ρ()=(a;  b; t;  c). 


Next we prove Theorem <ref>.  As we shall see the proof is completely constructive in the sense that it gives an explicit way of constructing a multicurve in N_g,n in finite number of steps. 



       

Let L be a minimal representative of ∈𝔏_g,n with generalized Dynnikov coordinates ρ()=(a;  b; t;  c).  Note that 2|a_i|=|α_2i-1- α_2i| and  2|t_i|=|γ_2i-1- γ_2i| give  the difference between below and above components in S_i and S'_i respectively by Lemma <ref>. Also |b_i| gives the number of loop components in S_i and S'_i  by Lemma <ref>. Let  m_i and n_i be the smaller of above and below components of L∩ S_i and L∩ S'_i respectively. From  Figure <ref> and Figure <ref> it is straightforward to compute β_i and β_n+i-1:

 


For 1≤ i≤ n-1,


 

    β_i    = {[       2m_i+2|a_i|                 ;; 2m_i+2|a_i|+ 2b_i                 . ].


For 1≤ i≤ g-1

     
    β_n+i-1   = {[           2n_i+|t_i| +ψ_i                         ;; 2n_i+|t_i|+ ψ_i +2b_n+i-1                         . ].






from which we get 

    β_i   =2[|a_i |+max(b_i,0)+m_i]
    β_n+i-1   =|t_i|+ψ_i +2[max(b_n+i-1, 0)+n_i]
  

Since β_n+i-1 is even from Remark <ref>, equality (<ref>) implies that |t_i|+ψ_i should be even. That is, |t_i|+max(c_i-|b_n+i-1|,0) is even by Lemma <ref>. 





Now, consider a subarc of L which intersects the last crosscap exactly once, has zero intersection with the other  crosscaps and intersects the horizontal diameter of the surface only between the  first puncture and the boundary exactly once as shown in Figure <ref>. Each such arc intersects each β_i and γ_i twice, and each α_i exactly once. We say that such arcs are almost boundary parallel, and write R for the number of almost boundary parallel arcs. 







 Using  β_i = β_1 - 2∑^i-1_j=1 b_j (1≤ i ≤ n+g-1) and  subtracting 2R from both sides of equation <ref> and equation <ref> we get


    β_1-2R   =2[|a_i |+max(b_i,0)+m_i-R+∑^i-1_j=1b_j ]  for   1≤ i ≤ n-1
    β_1-2R   =2[max(b_n+i-1, 0)+n_i-R+∑^n+i-2_j=1b_j]+|t_i |+ψ_i   for   1≤ i ≤ g-1.



One crucial fact is that m_i-R=0  for some 1≤ i≤ n-1 or n_i-R=0 for some   1≤ i≤ g-1 since otherwise there would be both above and below components in each of the S_i and S'_i except for those which arise from almost boundary arcs, but this would mean L contains  boundary parallel curves which is impossible. Then,
 
When m_i-R=0;



    β_1-2R=2[|a_i |+max(b_i,0)+∑^i-1_j=1 b_j ],



When m_i-R>0;



    β_1-2R>2[|a_i |+max(b_i,0)+∑^i-1_j=1 b_j],


When n_i-R=0;



    β_1-2R=2[max(b_n+i-1, 0)+∑^n+i-2_j=1b_j]+|t_i |+ψ_i,



When n_i-R>0;



    β_1-2R>2[max(b_n+i-1, 0)+∑^n+i-2_j=1b_j]+|t_i |+ψ_i.



Therefore, setting

    X_i   =2[|a_i |+max(b_i,0)+∑^i-1_j=1 b_j]
    
    Y_i   =2[max(b_n+i-1, 0)+∑^n+i-2_j=1b_j]+|t_i |+ψ_i

we get 


    β_1-2R= max_1≤ s ≤ n-1
    
                      1≤ k≤ g-1{X_s,Y_k}

and hence 


    β_i-2R= max_1≤ s ≤ n-1
    
                      1≤ t≤ g-1{X_s,Y_k}-2∑^i-1_j=1b_j

as required. Next, we compute R. Let




 
    Z_i=max_1≤ s ≤ n-1
    
                      1≤ k≤ g-1{X_s,Y_k}-2∑^i-1_j=1b_j

 By (<ref>), β_n+g-1=Z_n+g-1+2R.  Also by Remark <ref> β_n+g-1=2c_g+2λ_g. It follows that when Z_n+g-1>2c_g we have 
β_n+g-1>2c_g and hence λ_g≠ 0. Therefore, R=0 since almost  boundary parallel arcs and non-core loop components of L∩ S'_g cannot exist  at the same time, and when Z_n+g-1< 2c_g we have Z_n+g-1< β_n+g-1 and hence R>0 and λ_g=0 which implies β_n+g-1=Z_n+g-1+2R=2c_g that is  R=c_g-Z_n+g-1/2. Therefore, R=max(0,c_g-Z_n+g-1/2). 





To compute α_i and γ_i we make use of the equalities in Lemma <ref>:

    max(β_i, β_i+1)   =α_2i-1+α_2i



    max(β_n+i-1, β_n+i)   =γ_2i-1+γ_2i/2-ψ_i


Since  2a_i= α_2i-α_2i-1 (1≤ i≤ n-1) we get from <ref> that


If β_i≥β_i+1 (i.e. b_i≥ 0)



    α_2i= a_i+β_i/2;   and  α_2i-1=-a_i+β_i/2.


If β_i+1≥β_i (i.e. b_i≤ 0)


    α_2i= a_i+β_i+1/2;    and  α_2i-1=-a_i+β_i+1/2.



That is to say:

    α_i   = {[  (-1)^ia _⌈ i/2 ⌉+β_⌈ i/2⌉/2                            ;; (-1)^i a_⌈ i/2⌉+β_1+⌈ i/2⌉/2                              ].
 

Similarly, since  2t_i= γ_2i-γ_2i-1  for each 1≤ i≤ g-1 we get from <ref> that



If β_n+i-1≥β_n+i (i.e. b_n+i-1≥ 0)



    γ_2i= t_i+β_n+i-1+ψ_i;   and  γ_2i-1=-t_i+β_n+i-1+ψ_i,


If β_n+i-1≤β_n+i (i.e. b_n+i-1≥ 0)




    γ_2i= t_i+β_n+i+ψ_i;   and  γ_2i-1=-t_i+β_n+i+ψ_i.


That is to say:

    γ_i        = {[ (-1)^i t_⌈ i/2 ⌉+β_n+⌈ i/2⌉-1+ψ_⌈ i/2 ⌉                                        ;    (-1)^i t_⌈ i/2⌉+β_n+⌈ i/2⌉+ψ_⌈ i/2 ⌉                                         ].
 as required.



We note that generalized Dynnikov coordinates for multicurves can be extended in a natural way to  generalized Dynnikov coordinates of measured foliations when the space of measured foliations on N_g,n is endowed with its usual topology.



 §.§ Generalized Dynnikov Coordinates on N_g,0


Let N_g,0 be the standard model of a non–orientable surface of genus g with one boundary component as shown in Figure <ref> and denote by 𝔏_g,0 the set of multicurves on N_g,0.  Let
𝒮_g,n={t; b;  c)∈ (ℤ^2(g-2)×ℤ^g)∖{0}: |t_i|+ψ_i   is even   1≤ i≤ g-2}. Let the function ρ𝔏_g,0→𝒮_g,0  be defined by 

    ρ(ℒ) = ( t; b;  c):=(t_1,…, t_g-2;  b_1, …, b_g-2;  c_1,…, c_g)
 

where 

    t_i   = γ_2i-γ_2i-1/2; 1≤ i≤ g-2, 
    
    b_i   =β_i-β_i+1/2; 1≤ i≤ g-2






  









For 1≤ i≤ g-1, ψ_i, λ_i  and λ_c_i are as given in Lemma <ref>. For i=1 we have  ψ_1=0,  λ_1=β_1/2-c_1 and
λ_c_1=c_1. Similarly, for i=g we have  ψ_g=0 , λ_g=β_g-1/2-c_g and λ_c_g=c_g as each component of L∩ S'_0 and L∩ S'_g intersecting the core curves should be core loop components.



The inverse of the coordinate function ρ is described similarly.  However, we need to extend the definition of almost boundary parallel arcs as they could also arise from the first crosscap as shown in Figure <ref>. That is,  an almost boundary parallel arc associated with crosscap g is a subarc of L which intersects crosscap g exactly once, has zero intersection with crosscaps 2 through g-1;  and intersects  either crosscap 1 or the diameter between crosscap 1 and the boundary exactly once as shown in Figure <ref>(a) and Figure <ref>(b). An almost boundary parallel arc associated with crosscap 1 is described similarly.   



We write R_1 and R_g for the number of almost boundary parallel arcs associated with crosscap 1 and crosscap g  respectively.  Observe from Figure <ref> that there are R=max(R_1, R_g) almost boundary components in total.  By the same argument as in the proof of Theorem  <ref> we have  β_i=Z_i+2R where


    Z_i=max_1≤ s≤ g-1{2max(b_s, 0)+ ψ_s +|t_s |+2∑^s-1_j=1b_j}- ∑^i-1_j=1b_j




 as there are no a_i coordinates. Then, β_1-2R= Z_1 and  β_g-1-2R= Z_g-1.   We have R_1=max(0, c_1-Z_1/2) and R_g=max(0, c_g-Z_g-1/2) yielding R=max(0, c_1-Z_1/2, c_g-Z_g-1/2 ). The computation of intersection numbers on the arcs γ_i is as in the proof of Theorem  <ref>. Therefore we get Theorem <ref> where ⌈ x ⌉ again denotes the smallest integer which is not less than x. 



 
 Note the special case g=2 where there are no t_i and  b_i coordinates. 
 






	
Let (t; b;  c)∈𝒮_g,0. Then (t;  b;  c) corresponds to a unique multicurve in ∈_g, 0 which has 


    γ_i     = {[ (-1)^i t_⌈ i/2 ⌉+β_n+⌈ i/2⌉-1+ψ_⌈ i/2 ⌉                                        ;    (-1)^i t_⌈ i/2⌉+β_n+⌈ i/2⌉+ψ_⌈ i/2 ⌉                                         ].
    β_i   = Z_i+2max(0, c_g-Z_g-1/2, c_1-Z_1/2)

 
 where

    Z_i=max_1≤ s≤ g-1{2max(b_s, 0)+ ψ_s +|t_s |+2∑^s-1_j=1b_j}- ∑^i-1_j=1b_j

 
	
	


Let ∈𝔏_g,0 have generalized Dynnikov coordinates ( t; b;  c).  Let ( t'; b';  c') and  (t”; b”;  c”) be the generalized Dynnikov coordinates of u_i() and u^-1_i() respectively. Then   (c'_i-1, c'_i)=(c_i,c_i-1) and (c”_i-1, c”_i)=(c_i, c_i-1) for 1≤ i≤ g; and for 1<i≤ g-2 (b'; t') and  (b”; t”') are as given in equations (<ref>) replacing the subscript i with i-1; for i=g-1 t_g-2', b_g-2', t_g-2”, b_g-2” are as given in equations (<ref>) replacing the subscripts g-1 and n+g-2 with g-2; and for i=1 we have



    t_1'    =[t_1B_1/f_0(1+B_1)+ t_1]
    
       t_1”   = [f̅_̅0̅(t_1(B_1+1)+t_1)/B_1]      
      B_1'    = [B_1e_0(1+t'_1)/t_1'1/(1+t_1)]
    
      B_1”   = [e̅_0B_1(1+t”_1)/t”_1t_1/(1+t_1)]

In all other cases t'_j=t_j, t”_j=t_j and b'_j=b”_j=b_j. 







§ ACTION OF CROSSCAP TRANSPOSITIONS

The goal of this section is to prove Theorem <ref> which describes  how generalized Dynnikov coordinates change under the action of  u_i and u^-1_i (1≤ i≤ g-1). The key ingredient for the derivation of the formulae in Theorem <ref> is a set of equalities associated with particular arc systems which we call clovers and scales.  These equalities are given in Lemma <ref>, Lemma <ref>, Lemma <ref> and Lemma <ref>.




 §.§ Clovers and Scales


A clover and a scale  about crosscaps ⊗_i and ⊗_i+1   are  two different configurations consisting of  two vertices v_1,  v_2 at ∂ N_g,n (identified to the puncture at ∞), five arcs T_1, T_2, T_3, T_4, T_5 with end points  v_1,  v_2 and a curve _i such that the  teardrops T_1 and T_2 encircle ⊗_i, the teardrops T_3 and T_4 encircle ⊗_i+1, T_5 joins v_1 and v_2; and  _i  is the essential simple closed curve bounding crosscaps i and i+1 as shown in Figure <ref>.  We say that the clover has leaves T_1, T_2, T_3 and T_4;  diagonal arc T_5 and diagonal curve _i. Also,  the scale has leaves T_1, T_4, T_5 and _i; and diagonal teardrops T_2 and T_3.  





         To compute the action of u_i and its inverse u^-1_i (1≤ i≤ g-1) in terms of generalized Dynnikov coordinates we will make use of certain equalities associated with clovers and scales in S'_i∪ S'_i+1, which we shall call clover and scale equalities throughout.   We shall consider  three types of clovers and four types of scales to obtain clover and scale equalities. These clovers and scales are depicted in Figure <ref> and  Figure <ref> respectively.





























































Clover and scale equalities can be considered as a generalization of a well known equation commonly known as the flip move which lets us change coordinates from one triangulation to another on punctured orientable surfaces <cit.>. 
Namely, if Q is a rectangle in a surface S and X_1, X_2, X_3, X_4, X_5 and X_6 are the number of intersections  on the four edges and the diagonals of Q with all of its corners at the punctures and containing no punctures in its interior and X_ij denotes the number of arcs joining edge X_i to X_j then there are two possibilities: either X_12 or X_34 is zero since curves are non intersecting. This yields the well known equation  
    max(X_1+X_2, X_3+X_4)=X_5+X_6.
   The method here will be similar and use case by case analysis for components of L∩(S'_i∪ S'_i+1) intersecting the clovers and the scales. 

In what follows we will again denote by L a minimal representative of a multicurve ∈𝔏_g,n with ρ()=(a, b; t, c) and intersection numbers (α, β; γ,  c).






  §.§.§ Components of L∩(S'_i∪ S'_i+1)

We can list all topological possibilities for connected components of  L∩ (S'_i∪ S'_i+1) (1≤ i ≤ g-1), up to isotopy, making use of their intersections with the core curves c_i and c_i+1,  and  the arcs γ_l (2i-1≤ l≤ 2i+1) and β_l (n+l-1≤ l≤ n+l+1) (and hence from their generalized Dynnikov coordinates). Let ⊗_i and ⊗_i+1 denote crosscap i and crosscap i+1 respectively.   Given a connected component X of  L∩ (S'_i∪ S'_i+1) we associate with it a  signature vector v_i=(c_i, c_i+1; b_n+i-1, b_n+i)∈ℤ_≥0^2×ℤ^2 such that  c_i and c_i+1; give the number of intersections between X and the  core curves of ⊗_i and ⊗_i+1 respectively, and for j=i, i+1


    b_j=β_n+j-1-β_n+j/2


where β_k  denote the number of intersections of X with β_k (k=n+j-1, n+j).
 Note that each signature vector v_i must  satisfy equation (<ref>) of Lemma <ref>, and the inequality ( b_n+i-1)( b_n+i)≤ 0 since  X is a connected component of  L∩ (S'_i∪ S'_i+1). Then each connected component of L∩ (S'_i∪ S'_i+1) is either a simple closed curve  supported in S'_i∪ S'_i+1, or  one of the following arcs described in Notation <ref>. 


 


Let 2i-1 ≤ l≤  m ≤  2i+2 and k=i, i+1.  


  1. X^v_i_lm:
 

  (a) If (l, m)≠ (2i-1,2i), (2i+1,2i+2) it passes above (resp. below) ⊗_i if l=2i-1 (resp. l=2i), and it passes above (resp. below) ⊗_i+1 if m=2i+1 (resp. m=2i+2). It has one end point on  β_n+i-1 and the other on  β_n+i+1.


  (b)  If (l, m)= (2i-1, 2i)  it passes both above  and below ⊗_i, and it has both end points on β_n+i-1. The case  (l, m)= (2i+1, 2i+2) is described similarly. 


  (c)  If l=m  it passes above (resp. below) ⊗_i if l=2i-1 (resp. l=2i)  and has both end points on β_n+i-1. The case  l=m=2i+1 is described similarly. 



 
  2.X^v_i_l; k: it passes above (resp. below) ⊗_i if l=2i-1 (resp. l=2i). It has  ψ_i=1 and both end points on  β_n+i-1. The cases l=2i+1 and l=2i+2 are described similarly. 


  3.  X_i,  X_i^v_i,  X_i+1, X_i+1^v_i and X_ii+1:   none of these components pass above or below ⊗_i and ⊗_i+1.  X_i and X_i^v_i have both end points on β_n+i-1 with ψ_i=0 and ψ_i≠ 0  respectively.  X_i+1 and X_i+1^v_i are described similarly. X_ii+1 has one end point on  β_n+i-1 and the other on  β_n+i+1.


See for example the arcs 5,15,16 for item 1(a); 2, 9, 14 for item 1(b);  7, 12 for item 1(c);  1,4, 6, 8  for item 2; and 3, 10,11,13 for item 3.  in Figure <ref>.
 




We shall omit the superscript v_i for when  b_n+i-1=b_n+i=0. See for example the arcs 3, 4, 5, 8, 15 and 16 in Figure <ref>. We write  [P_i] for the set of simple closed curves in S'_i∪ S'_i+1, and  [X^v_i_lm], [X^v_i_l; k] and [X_i] for  the set of arcs described in item 1, item 2. and item 3 respectively. 
		





Here and in what follows we write L_i to denote the arc system L∩(S'_i∪ S'_i+1) for convenience.

 

 Let X be a component  of L_i with signature vector v_i=(c_i, c_i+1; b_n+i-1, b_n+i).  We say that X is a standard arc if at least one of b_n+i-1 and b_n+i equals zero (Figure <ref>). We say that X is twisted if  b_n+i-1<0  and b_n+i>0. 
 

 
   














 
 
 We say that a component of L_i is a positive arc if it satisfies t_i-t_i+1>0, it is negative if  t_i-t_i+1<0 and that it is neutral if t_i+1-t_i=0. 
 
Figure <ref> depicts examples for negative and neutral standard components of L_i. Other standard components can be obtained by symmetry, reflecting them in the arc  β_n+i, or the diameter of the surface. 

 
 Since b_n+i-1<0  and b_n+i>0, and t_i-t_i+1=0 for a simple closed curve in S'_i∪ S'_i+1 we regard each such curve neutral and twisted. 




A twisted component of L_i is called a  negatively (resp. positively) half twisted  arc if it is the u^-1_i  (resp. u_i)  image of a standard arc.  A negative (resp. positive)  twisted component of L_i which is not half twisted is called negatively (resp. positively) highly twisted. Each simple closed curve and twisted neutral component of L_i is called neutrally twisted.  

 


 
  Let λ_k denote the number of non–core loop components of L∩ S'_k (k=i, i+1). We denote



    λ^+_i+1= λ_i+1   if  b_n+i >0
    
                 0    if  b_n+i <0
             and  λ^-_i= λ_i   if  b_n+i-1 <0
    
                 0    if  b_n+i-1 >0

We describe λ^+_c_i+1 and λ^-_c_i similarly. Elsewhere x^+ denotes max(x, 0). 

 

  Let A'_i and B'_i be the number of above and below components of L_i given in Lemma <ref>.


Let  A'_i,i+1 = min(A'_i, A'_i+1) and B'_i,i+1 = min(B'_i, B'_i+1).
We write 
 
    Δ_i(A)=A'_i-A'_i,i+1    and    Δ_i+1(A)=A'_i+1-A'_i,i+1
    Δ_i(B)=B'_i-B'_i,i+1    and    Δ_i+1(B)=B'_i+1-B'_i,i+1

 

 


 

Geometrically, A'_i, i+1 and B'_i,i+1 give the number of components of L_i which  lie entirely above and below the diameter of the surface respectively. Therefore, A'_i, i+1=X_2i-12i+1 and B'_i, i+1=X_2i2i+2. Then, Δ_i(A) (resp. Δ_i(B)) is the number of above (resp. below) components of L∩ S'_i which are not  contained in the arcs X_2i-1, 2i+1 (resp.  X_2i, 2i+2).    Δ_i+1(A) and  Δ_i+1(B) are interpreted similarly. 

Figure <ref> illustrates these parameters which we will refer to later to describe other parameters. 





The condition t_i-t_i+1>0 in Definition <ref>  implies that either Δ_i(B)≠ 0 or Δ_i+1(A)≠ 0 holds for a positive component. Then,  X^v_i_lm is positive  if l=2i+1 or m=2i. Similarly, X^v_i_l; k (k=i, i+1) is positive if l=2i+1 or l=2i. Similar arguments hold for negative components. Similarly, the condition t_i+1-t_i=0 implies that a neutral component in item 1(a) (i.e. X_2i-12i+1 and X_2i2i+2)  and  item 2. has Δ_k(A)=Δ_k(B)=0 (k=i, i+1). A neutral component in item 1(b) has either Δ_i(A)≠ 0 and Δ_i(B)≠ 0 (i.e. X^v_i_2i-12i)  or Δ_i+1(A)≠ 0 and Δ_i+1(B)≠ 0  (i.e. X^v_i_2i+12i+2). 





 We say that two arcs in S'_i∪ S'_i+1 are compatible if they can be embedded disjointly in  S'_i∪ S'_i+1.   
 

A positive and a negative arc are compatible only when they form the arc systems scissors,  anchors or  ribbons. 
   


      
  Scissors at ⊗_i consists of the arcs  X_2i+2;  i  and X_2i+1;  i (Figure  <ref>(a)), and scissors at  ⊗_i+1 consists of the arcs X_2i;  i+1 and  X_2i-1;  i+1. 
 
 A left anchor  consists of  X^(1,1; -1,0)_2i+2,i+1  and X^(1,1; -1,0)_2i+1,i+1 (see Figure  <ref>(b)), and a right anchor  consists of X^(1,1; 0,1)_2i; i and X^(1,1; 0,1)_2i-1; i.
   
 			A  left ribbon  consists of arcs from the sets [X_2i+2;  i] and [X_2i+1; i] (arcs of a ribbon are twisted and can have different signature vectors). And a right ribbon   consists of elements from the sets [X_2i; i+1] and [X_2i-1; i+1].  Note that scissors, ribbons and anchors may contain multiple copies of the same arc.
			
			The positive and negative arcs of given scissors are respectively called positive and negative arms for the scissors. For example,   X_2i+2;  i and X_2i+1;  i are  negative and positive arms of the scissors respectively. Positive and  negative arms of anchors and ribbons are described similarly. 




     
    Note that scissors, anchors and ribbons are not compatible with each other.  In fact if a component of L_i is compatible with any  scissors , anchors or ribbons it must be a neutral arc. 
 






If L_i  is comprised of standard arcs it is called standard, and if it contains at least one twisted arc it is called  twisted. If it contains positive arcs possibly with neutral arcs it  is called positive. The case when L_i is negative is described similarly. If L_i contains only neutral arcs it is called neutral, and if it contains both positive and negative arcs it is called mixed.      

Therefore L_i is mixed if  and only if it contains either scissors or an anchor or a ribbon, and that the only case L_i is both twisted and mixed is when it contains ribbons. 
 
   
         

  §.§.§ Clover Equalities

          Let  u^-1_i(γ; β)=(γ'; β') and u_i(γ; β)=(γ”; β”).   A clover of type I has leaves γ_2i-1, γ_2i, γ_2i+1, γ_2i+2, the diagonal arc β_n+i  and the diagonal curve _i  (Figure <ref>(a)).    A clover of type II and a clover of type III are the images of a clover of type I under the mapping classes u^-1_i and u_i respectively, and hence are as depicted in Figure <ref>(b) and Figure  <ref>(c) respectively.

         


         
          We present equalities associated with a clover of type I, type II and type III given in Lemma <ref>,  Lemma <ref> and Lemma <ref> respectively. Here and in what follows we abuse notation again using the symbols in Notation <ref> to denote the number of corresponding components  of L_i, and the symbols γ; β, γ'; β', γ”; β” to denote the number of intersections.  First, we  fix some notation which will be necessary in the proof of Lemma <ref>. 
          
           



Given scissors at ⊗_i and scissors at ⊗_i+1 let s_i=min(X_2i+1;  i, X_2i+2;  i) and s_i+1=min(X_2i-1; i+1, X_2i;  i+1) respectively.  Given a right and a left anchor in   S'_i∪ S'_i+1  let  z_i=min(X^(1,1; 0,1)_2i-1, i, X^(1,1;0,1)_2i, i) and z_i+1=min(X^(1,1; -1,0)_2i+1, i+1, X^(1,1; -1,0)_2i+2, i+1) respectively. Given a left and a right ribbon in S'_i∪ S'_i+1  let r_i=min(X^v_i_2i+1; i, X^v'_i_2i+2; i) and  r_i+1=min(X^v_i_2i-1; i+1, X^v'_i_2i; i+1) respectively. Finally, let  N_i and N_i+1 denote  the sum of all neutral  arcs from the sets [X_2i+12i+2] and [X_2i-12i] respectively described in 1(b) in Notations <ref> except for the standard components X^(0,0;0,1)_2i-12i and X^(0,0;-1,0)_2i+12i (i.e. those with zero intersection with the crosscaps).  For each 1≤  i≤ g-1 we set  d_i=ϵ_i+ϵ_i+1 where  ϵ_i=s_i+z_i+1+w_i+N_i and ϵ_i+1=s_i+1+z_i+w_i+1+N_i+1. 
 
 
 
 If ϵ_i≠ 0 then ϵ_i+1=0, and that only one of s_i, z_i+1 and w_i can be different than zero by Remark <ref>. Similarly,  if ϵ_i+1≠ 0 then ϵ_i=0 and only one of s_i+1, z_i and w_i+1 is  nonzero.
 
 



 
Given a clover of type I in S'_i∪ S'_i+1 we have

    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2d_i.

 where d_i is  as given  in Notation <ref>.

  
  Let [X_2i-12i]={X^v_i_2i-12i},  [X_2i+12i+2]={X^v_i_2i+12i+2} and  [X_i] be the sets of neutral components of L_i described in item 1(b) and item 3.  of Notation <ref> respectively.  For 2i-1 ≤ l≤  m ≤  2i+2 with (l, m)≠ (2i-1,2i), (2i+1,2i+2) and k∈{i, i+1} write [X_l]={X^v_i_l, m,   X^v_i_l; k} for  the set of components described in item 1(a), item 1(c) and item 2. of Notation <ref>, and  [P_i] for set of simple closed curves supported in S'_i∪ S'_i+1.  By Remark <ref> we have the following cases:


  Case 1:  L_i is either positive or negative  or neutral.  





  a) [X_2i-1]∪ [X_2i+2] ≠∅ and [X_2i]∪ [X_2i+1]=∅; 

  b) [X_2i-1]∪ [X_2i+2] = ∅ and [X_2i]∪ [X_2i+1]≠∅;

  c) [X_2i-1]∪ [X_2i+2] = ∅ and [X_2i]∪ [X_2i+1]= ∅;



  Case 2: L_i is mixed that is L_i contains  either scissors or an anchor or a ribbon. There are 2 subcases 





  a) [X_2i+1]∪ [X_2i+2] ≠∅ and [X_2i-1]∪ [X_2i]=∅; 

  b) [X_2i-1]∪ [X_2i+2] = ∅ and [X_2i]∪ [X_2i+1]≠∅;


 
  
  		 For Case 1 we have s_k=z_k=r_k=0 (k=i, i+1)  since L_i is not mixed.    Suppose first that Case 1a) holds. Then,  L_i is negative by Remark <ref>  and Definition <ref> (see for example Figure <ref>(i)); and we have γ_2i-1+γ_2i+2> γ_2i+γ_2i+1.   First assume that L_i is standard.   It is easy to check that each negative standard component in [X_2i-1]∪ [X_2i+2], each curve in [P_i]  and each neutral component except for X^(0,1;0,1)_2i-12i and  X^(0,1;0,1)_2i+12i+2  satisfies  
    2β_n+i+C_i=γ_2i-1+γ_2i+2=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1).

 Suppose that  X^(0,1;0,1)_2i-12i≠ 0 and hence X^(0,1;0,1)_2i+12i+2= 0 (note that elements of [X_2i-12i] and  [X_2i+12i+2]  are not compatible).                                    We check that X^(0,1;0,1)_2i-12i satisfies  γ_2i-1+γ_2i+2=γ_2i+γ_2i+1, and  
    2β_n+i+C_i-(γ_2i-1+γ_2i+2)=2X^(0,1;0,1)_2i-12i.
 That is, 
    
    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2 X^(0,1;0,1)_2i-12i.

    
Similar argument holds for the case X^(0,1;0,1)_2i-12i=0 and X^(1,0;-1,0)_2i+12i+2≠ 0, and we get    
 
    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2 X^(1,0;-1,0)_2i+12i+2


  


 	 Equalities (<ref>),  (<ref>) and   (<ref>)  are also satisfied for  corresponding twisted components: given a standard component X_l, m of L_i with intersection numbers  γ_j (2i-1≤ j≤ 2i+2), β_n+i and _i the twisted component X^v_i_l, m  has the same  number of intersections on _i and increases the number of intersections on each γ_j (2i-1≤ j≤ 2i+2) and 2β_n+i (and hence 2β_n+i+C_i) by the same amount determined by its signature vector (Figure <ref>(ii)). Therefore  equality (<ref>) is also satisfied for each X^v_i_l, m. Similar argument holds for other twisted components. Then we have d_i=ϵ_i=N_i, and 

	 
  

  
    
    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2d_i
 
 as required.   Case 1b)  follows by symmetry.  For Case 1c), we note that since L_i consists of only neutral components and each neutral component satisfies γ_2i-1+γ_2i+2= γ_2i+γ_2i+1 except for X^(0,1;0,1)_2i-12i and  X^(0,1;0,1)_2i+12i+2 as shown above, we get either ϵ_i=N_i≠ 0 or  ϵ_i+1=N_i+1≠ 0 since X^(0,1;0,1)_2i-12i and  X^(0,1;0,1)_2i+12i+2 are not compatible. Therefore, d_i=ϵ_i+ϵ_i+1, and
 
 
    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2 d_i.

 as required.
    
     








Case 2a) is divided into 3 subcases. Either L_i contains scissors at ⊗_i or a left anchor or a left ribbon  by Definition <ref> and Remark <ref>. Assume that  L_i contains scissors at ⊗_i. Then  we have X_2i+1; i≠ 0, X_2i+2; i≠ 0 and hence d_i=ϵ_i=s_i=min(X_2i+1;  i, X_2i+2;  i)+N_i where N_i= X^(1,0;-1,0)_2i+12i+2 (twisted neutral components are not compatible with scissors).  We obtain




    2β_n+i+C_i-2X_2i+1; i-2X^(1,0;-1,0)_2i+12i+2   =γ_2i-1+γ_2i+2



    2β_n+i+C_i-2X_2i+2; i-2X^(1,0;-1,0)_2i+12i+2   =γ_2i+γ_2i+1

 as shown in Figure <ref>(i).  Observe that γ_2i-1+γ_2i+2≥γ_2i+γ_2i+1 if and only if X_2i+2; i≥ X_2i+1; i. Then from equation (<ref>) and equation (<ref>)  we get


    2β_n+i+C_i   =max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2min(X_2i+2; i, X_2i+1; i)+2X^(1,0;-1,0)_2i+12i+2
       =max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1)+2s_i+2X^(1,0;-1,0)_2i+12i+2.


as required. Similarly, if L_i contains a left anchor it contains no scissors, no ribbons and no right anchor.  We have d_i=z_i+1+N_i and equation (<ref>) is verified analogously. The case when there is a left ribbon is proved similarly.  Case 2b) is also divided into 3 subcases: Either  L_i contains scissors at ⊗_i+1 or a right anchor or a right ribbon by Definition <ref> and Remark <ref>.  Then Case 2b)  follows immediately from Case 2a) by symmetry.  










 

The proof of Lemma <ref> shows that each component of L_i except for certain arcs and arc systems satisfies

    2β_n+i+C_i=max(γ_2i-1+γ_2i+2, γ_2i+γ_2i+1).
 We shall call these arcs and arc systems exceptional arcs and arc systems with respect to a clover type I; and d_i the exceptional parameter for a clover of type I. 

Each neutral arc in [X_2i+12i+2]∖{X^(0,0;-1,0)_2i+12i+2} and [X_2i-12i]∖{X^(0,0;0,1)_2i-12i}  is called an exceptional arc with respect to a clover type I at ⊗_i and ⊗_i+1 respectively (for example in Figure <ref> the arc labeled 2 is an exceptional arc at ⊗_i while the arc labeled 9 is not); scissors, ribbons and anchors  in S'_i∪ S'_i+1 are called exceptional arc systems with respect to a clover of type I.


 
 Let Δ_k(A) and Δ_k(B) (k=i, i+1) be as described in Notation  <ref>. The proof of Lemma <ref> follows immediately from the definition of exceptional arcs and arc systems.
 

 Δ_i+1(A)≠ 0 and Δ_i+1(B)≠ 0 if and only if there exists at least one of the following arc or arc systems in S'_i∪ S'_i+1:  an arc from the set [X_2i+12i+2], scissors at ⊗_i, a left anchor or a left ribbon.  Similarly, Δ_i(A)≠ 0 and Δ_i(B)≠ 0 if and only if at least one of the following exists in S'_i∪ S'_i+1: an arc from the set [X_2i-12i], scissors at ⊗_i+1, a right anchor or a right ribbon. 


Then, we can compute the exceptional parameter d_i=ϵ_i+ϵ_i+1 in terms of generalized Dynnikov coordinates.  We first give the following preliminary lemma.


Let X_2i+12i+2^(0,0;-1,0) and X_2i-12i^(0,0;0,1)  be as given in Notation <ref>. Then,
 
    X_2i+12i+2^(0,0;-1,0)   =min(Δ_i+1(A),  Δ_i+1(B), λ^-_i)
    
     X_2i-12i^(0,0;0,1)   =min(Δ_i(A),  Δ_i(B), λ^+_i+1)




The proof follows from Remark <ref> and Figure <ref>. 









  Let ϵ_k (k=i, i+1) be as described in  Notation <ref>. Then d_i=ϵ_i+ϵ_i+1 where


    ϵ_i   =min(Δ_i+1(A), Δ_i+1(B))-X_2i+12i+2^(0,0;-1,0)
    ϵ_i+1   =min(Δ_i(A), Δ_i(B))-X_2i-12i^(0,0;0,1)

and X_2i-12i^(0,0;0,1) and X_2i+12i+2^(0,0;-1,0) are as given in Lemma <ref>.



We compute ϵ_i. ϵ_i+1  is computed analogously reflecting in the arc β_n+i. If at least one of Δ_i+1(A) and Δ_i+1(B) is equal to zero then ϵ_i=0 by Lemma <ref>. So suppose that Δ_i+1(A)≠ 0 and Δ_i+1(B)≠ 0.  

 


  Case 1. There exists no exceptional arc system with respect to a clover type I  in which case there must be exceptional arcs  from the set [X_2i+12i+2].




  Case 2.  There exists an exceptional arc system with respect to a clover type I which can be either scissors at ⊗_i or a left anchor or a left ribbon possibly with compatible exceptional arcs  from the set [X_2i+12i+2]. 

 

We have two subcases in Case 1: (a) Δ_i+1(A)≤Δ_i+1(B) and (b) Δ_i+1(A)≥Δ_i+1(B). Assume that we are in Case 1(a). Then there exist negative but no positive arcs in S'_i∪ S'_i+1  since only negative components increase the difference Δ_i+1(B)-Δ_i+1(A) by 
Remark <ref> and that there is no exceptional arc system with respect to a clover type I in  S'_i∪ S'_i+1  by assumption. Since each element of [X_2i+12i+2] increases Δ_i+1(A) and Δ_i+1(B) by 1 and N_i of those are exceptional we have


    Δ_i+1(A)   =N_i+X_2i+12i+2^(0,0;-1,0)≤Δ_i+1(B)


(recall that X_2i+12i+2^(0,0;-1,0) is the only element of [X_2i+12i+2] which is not exceptional). Since d_i=ϵ_i=N_i we get d_i=min(Δ_i+1(A), Δ_i+1(B))-X_2i+12i+2^(0,0;-1,0) as required. Case 1(b) is proved similarly. Now assume that  we are in Case 2.   Assume that there exists scissors at ⊗_i (Figure <ref>). Since the only element of [X_2i+12i+2]  which is compatible with the scissors is the standard exceptional arc X_2i+12i+2^(1,0;-1,0) we have N_i=X_2i+12i+2^(1,0;-1,0), and hence d_i=ϵ_i=s_i+N_i. By Remark <ref> every other arc compatible with the scissors is neutral and has no affect on Δ_i+1(A) and Δ_i+1(B). Therefore,


    Δ_i+1(A)   =X_2i+1; i+X_2i+12i+2^(1,0;-1,0)
    Δ_i+1(B)   =X_2i+2; i+X_2i+12i+2^(1,0;-1,0)

Since s_i=min(X_2i+1; i,X_2i+2; i),  N_i=X_2i+12i+2^(1,0;-1,0) and X_2i+12i+2^(0,0;-1,0)=0 we obtain  
    d_i=min(Δ_i+1(A), Δ_i+1(B))=min(Δ_i+1(A), Δ_i+1(B))+X_2i+12i+2^(0,0;-1,0)
 
as required. In the cases when there is a left anchor and a left ribbon we note that there may exist both twisted and standard exceptional arcs in  [X_2i+12i+2]. Also,  X_2i+12i+2^(0,0;-1,0)= 0 if there exists a left ribbon and X_2i+12i+2^(0,0;-1,0)≥ 0 if there exists a left anchor. Again, since each positive (resp. negative) arm of a left anchor or a left ribbon increases Δ_i+1(A) (resp. Δ_i+1(B)) by 1,  and only neutral arcs are compatible with exceptional arc and arc systems the proof follows similarly.   
	
	





  Taking the u_i^-1 images of exceptional arcs with respect to a clover type I in S'_i∪ S'_i+1 we obtain elements of [X_2i+2; i+1] and [X_2i-1; i] which are called exceptional arcs with respect to a clover of type II in S'_i∪ S'_i+1. Similarly, taking the u_i^-1 images of scissors, anchors and ribbons in S'_i∪ S'_i+1 we get negatively half twisted scissors, anchors and ribbons which are  exceptional arc systems with respect to a clover type II  (see Figure <ref> and Figure <ref> for some examples). This leads us to the equality in Lemma <ref>. But first we introduce some notation for the parameters associated with exceptional arc and arc systems with respect to a clover type II. 







 
Let e_i=ϵ'_i+ϵ'_i+1 where ϵ'_i=s'_i+z'_i+r_i'+N'_i where N'_i and N'_i+1  denote the number of exceptional arcs of type II which are from the sets [X_2i-1; i] and [X_2i+2; i] respectively, and 

    s'_i   =min(u^-1_i(X_2i+1; i), u_i^-1(X_2i+2; i)),    s'_i+1=min(u^-1_i(X_2i-1; i+1), u^-1_i(X_2i; i+1)),
      
    z'_i   =min(u^-1_i(X^(1,1; -1,0)_2i+1, i+1), u^-1_i(X^(1,1; -1,0)_2i+2, i+1)),   z'_i+1=min(u_i^-1(X^(1,1; 0,1)_2i-1, i), u_i^-1(X^(1,1;0,1)_2i, i)),
    
     r'_i   =min(u_i^-1(X^v_i_2i+1, i), u_i^-1(X^v'_i_2i+2, i)),      r'_i+1=min(u_i^-1(X^v_i_2i-1, i+1), u_i^-1(X^v'_i_2i, i+1)).

e_i is  called  the exceptional parameter for a clover of type II.


 
 
 
Given a clover of type II we have
 
    2β'_n+i+C_i=max(γ_2i+1+γ_2i, γ'_2i+1+γ'_2i)+2e_i.

where the exceptional parameter e_i is as given in Notation <ref>. 

We require the following parameters to compute e_i and the other exceptional parameters in terms of generalized Dynnikov coordinates. 



     Let X be a component  of L_i with v_i=(c_i, c_i+1; b_n+i-1, b_n+i).  Write
 
    χ_i(A)   =(b_n+i)^+-Δ_i(A),   χ_i+1(A)=(-b_n+i-1)^+-Δ_i+1(A)
    χ_i(B)   =(b_n+i)^+-Δ_i(B),  χ_i+1(B)=(-b_n+i-1)^+-Δ_i+1(B)
 

 
  
   We define χ(X) as follows: 
    χ(X)=min( χ_i(B), χ_i+1(A))   if  X  is positive
    min( χ_i(A), χ_i+1(B)),    if X is negative 
    min( χ_k(A), χ_k(B)),    if X is neutral
We note that  if X is neutral χ_k(A)=χ_k(B) for each k=i, i+1. 
  
Geometrically, χ(X)  gives information about the amount of twist of X, and reveals whether  X positive or negative or neutral by Remark <ref>. Observe that a standard component X of L_i either has χ(X)=0 or χ(X)=-1. The possibilities for the latter case is given in Remark <ref>.


Let X be standard. When X is   negative χ(X)=-1 if and only if 
    X ∈{X^(0,0;0,1)_2i-12i-1,  X^(0,1;0,1)_2i-12i-1,  X^(0,0; -1,0)_2i+22i+2,  X^(1,0;-1,0)_2i+22i+2,  X_2i-12i+2}
 See for instance l_2=X^(0,0;-1,0)_2i+22i+2,  l_3=X^(1,0;-1,0)_2i+22i+2,  l^⋆_3=X_2i-12i+2 in Figure <ref>. Similarly, when  X is   positive    χ(X)=-1 if and only if   
    X∈{X^(0,0; -1,0)_2i+12i+1,  X^(1,0;-1,0)_2i+12i+1,  X^(0,0; 0,1)_2i2i,  X^(0,1;0,1)_2i2i,  X_2i2i+1}.
   


Note that a standard arc cannot be compatible with a highly twisted arc X with χ(X)>1. Furthermore, if an arc X has ψ_i+1≠ 0 it is either a standard arc or a highly twisted arc from the sets [X_2i+2;i+1] or [X_2i+1;i+1]. Similar argument holds for an arc X with ψ_i≠ 0.  


Suppose that L_i is not mixed.  We define χ(i) for L_i by replacing X with L_i and removing all hats from the symbols given in Notation <ref>.     


Let X_2i-12i=X_2i-12i^(0,0;0,1) and X_2i+12i+2=X_2i+12i+2^(0,0;-1,0) be as given in Lemma <ref>. For notational simplicity we shall denote  Λ_i=λ^-_i-X_2i+12i+2, Λ_i+1= λ^+_i+1-X_2i-12i and

    Δ̅_i(B)   = Δ_i(B)-X_2i-12i,    Δ̅_i(A)= Δ_i(A)-X_2i-12i
    Δ̅_i+1(B)   = Δ_i+1(B)-X_2i+12i+2,   Δ̅_i+1(A)= Δ_i+1(A)-X_2i+12i+2

 We also introduce the following components for twisted components of L_i:
 
    p_1=2(-b_n+i-1)^+-Δ_i+1(B),    p_2=2(b_n+i)^+-Δ_i(A)
    
    p_3=2(-b_n+i-1)^+-Δ_i+1(A),    p_4=2(b_n+i)^+-Δ_i(B)

Geometrically, p_1 denotes the number of loop components of L∩ S'_i which are not contained in below components of L∩ S'_i+1 for a twisted component of L_i. The interpretation of the other parameters p_k is similar.

To understand these parameters better let us consider Figure <ref> again where L_i consists of three components: l_1=X_2i+12i+2,   l_2=X_2i+1, 2i+2 and 
l_3=X^(1,2; -1,1)_2i+2; i+1.  Since l_1 and l_2 are neutral χ(l_1)=χ(l_2)=0. Also, l_3 is a negative twisted component with  χ(l_3)=min(1, 1)=1 and 
p_1=3. 






Let L_i be negative. Then, e_i=ϵ'_i+ϵ'_i+1 where

    ϵ'_i=min(Λ_i, ψ_i+1,Δ̅_i+1(B), p_1)   and  ϵ'_i+1=min(Λ_i+1, ψ_i, Δ̅_i(A), p_2)







Here we only prove ϵ'_i.  The formula for ϵ'_i+1 is obtained similarly. We first note that each exceptional arc in [X_2i+2; i+1]  and negatively half twisted scissors at ⊗_i, left anchor and left ribbon increases each λ^-_i, ψ_i+1, Δ_i+1(B) by 1 (see Figure <ref> and  Figure <ref>).    Therefore if at least one of λ^-_i, ψ_i+1, Δ_i+1(B) equals zero for L_i  we get ϵ'_i=0.  For convenience, let us say that a subset of L_i has property P if it satisfies  λ^-_i≠ 0, ψ_i+1≠ 0 and  Δ_i+1(B)≠ 0. The proof is based on constructing all possible configurations of arcs (i.e. compatible sets) satisfying  property P, and verifying that equation (<ref>) holds for each such collection of arcs. Let L_i have property P. The constraint provided by property P and Remark  <ref> imply that each element of L_i belongs to one of the two sets described as follows: The first set I is the subset of L_i  whose elements affect none of  the values λ^-_i, ψ_i+1 and Δ_i+1(B) yet compatible with those satisfying property P. The second  set S contains negative components affecting at least one of λ^-_i, ψ_i+1, Δ_i+1(B). Furthermore, S is partitioned into two subsets S_0 and S_1 such that if X∈ S_0 then χ(X)≤ 0 and X is one of the following arcs depicted in Figure <ref>; and  if X∈ S_1 then χ(X)≥ 1 and X is compatible with an arc with ψ_i+1≠ 0. In particular, if  X∈ S_1 with ψ_i+1=0 it  has χ(X)=1 and it is one of the arcs l_15, l_16, l_17, l_18 depicted in Figure <ref>; and if ψ_i+1≠ 0 it is a highly twisted  exceptional arc from the set [X_2i+2;i+1] such as l_18 and l_19 as depicted in Figure <ref>.   Let 𝒞_k be the family of k–element subsets of L_i (i.e. possible configurations of exactly k arcs from the sets S_0 and S_1) satisfying property P. Again, by abuse of notation the symbols l_i we use to indicate these arcs will also denote the number of corresponding  arcs. 



 






 
	








First, observe from Figure <ref> and Figure <ref> that l_0 is a standard exceptional arc with respect to a clover type II.  Also, l_18, l_19 are examples for twisted exceptional arcs; and {l_1, l_9}, {l_4, l_8} and {l_12, l_15} form  exceptional arc systems of with respect to a clover type II. 








  Case 1.  χ(i)≤ 0: Each component of L_i∖ I belongs to S_0 where χ(l_2)= χ(l_3)=χ(l_⋆)=-1;  and χ(l_k)=0 for every other l_k in S_0. As to be explained later in Remark <ref>, we need another parameter p_1=2(-b_n+i-1)^+-Δ_i+1(B). For simplicity, we list the 4–tuples (λ_i, ψ_i+1, Δ_i+1(B), p_1)  in Table <ref> corresponding to the arcs l_k in S_0.  Observe from Table <ref> that l_0 is the only element satisfying property P alone hence 𝒞_1={{l_0}}. Furthermore, it  increases  each λ_i, ψ_i+1, Δ_i+1(B) and p_1 by 1, yielding ϵ'_i=N_i=l_0=min(M_1,   p_1) as required.  In order to construct 𝒞_k (k>1) we make use of another set I_k which is the set of k-tuples (i_1, i_2,…, i_k) for the compatible components l_i_1, l_i_2,…, l_i_k where i_j∈{⋆, k:0≤ k≤ 14}. We chose to use the star symbol here to indicate that l_⋆ is only arc that is not compatible with any exceptional arc or arc system (in fact it is compatible with only l_1, l_2 and l_3). We get,  I_2=I^(1)_2∪ I^(2)_2∪ I^(3)_2, where 
    I^(1)_2={(0, j): 1≤ j≤ 14},    I^(2)_2={(1,⋆), (1, j): 1< j< 14, j≠  7, 8}  and
  

    I^(3)_2=   {(2,3), (2,⋆), (2,7), (2,9), (2,14), (3,⋆), (3,4), (3, 7), (3, 8), (3,9), (3, 10), (3,14), (4, 6), (4, 8), 
       (4, 10), (4,12), (4,14),  (5, 6), (5,13), (5,14), (6,12), (6,13), (6,14), (7, 8), (7,9),(7, 10), (7,14), 
       (8, 9), (8, 10), (8,14), (9, 10), (9, 11), (10,12),(11,13), (12,13)}.

 and obtain that  𝒞_2=𝒞^(1)_2∪𝒞^(2)_2 where 
    𝒞^(1)_2={{l_0, l_j}: 1≤ j≤ 14}  and  𝒞^(2)_2={{l_1, l_2}, {l_1, l_9}, {l_4, l_8}, {l_4, l_14}, {l_5, l_14}, {l_6, l_14}}.
  




First consider 𝒞^(1)_2. Recall that to compute the parameters  associated with a compatible set we simply add the corresponding components of arcs. For example,  from Table <ref> we compute that (M_1, p_1)=(1,1) for (l_0, l_1).  We immediately check that for each {l_0, l_j}∈𝒞^(1)_2 we have l_0=N'_i=M_1≤ p_1, and  and therefore ϵ'_i=N'_i=l_0=min(M_1, p_1) (note that taking multiple copies of arcs does not change the formula).   We continue with 𝒞^(2)_2. The set {l_1, l_2} contains no exceptional arc or arc systems with respect to a clover type II, and observe from Table <ref> as above that p_1=0. Therefore,  ϵ'_i=0=min(M_1, p_1) as required.  Similarly, we check {l_1, l_9}. This set contains negatively half twisted scissors (Figure <ref>(a)) but no exceptional arcs hence we have ϵ'_i=s'_i. We check that l_1=ψ_i+1 and l_9=λ_i=Δ_i+1. Therefore, s'_i=M_1=min (l_1, l_9). Furthermore, p_1 is increased  by 1 by both l_1 and l_9. This implies M_1≤ p_1 yielding  ϵ'_i=s'_i=min(M_1, p_1) as required. Similarly,  {l_4, l_8} is a negatively half twisted left anchor (Figure <ref>(b))  and hence ϵ'_i=z'_i. We check that z'_i=min(l_4, l_8)=M_1≤ p_1 yielding ϵ'_i=z'_i=min(M_1, p_1) as required. Finally, none of (l_j, l_14)∈𝒞^(2)_2 contains an exceptional arc or arc system. Since l_14=X_2i+12i+2^(0,0;-1,0) we get M_1=0 yielding ϵ'_i=0 as expected.
The formula can be verified similarly for elements of 𝒞_3 as follows. 

		We have I_3=I^(1)_3∪ I^(2)_3∪ I^(3)_3 where I^(1)_3={(0, j, k): (j, k)∈ I_2, j, k≠⋆} and I^(2)_3={(1, j, k):  (j, k)∈ I_2, j, k∉{7, 8, 14}} and


    I^(3)_3={   (2,3,7), (2,3,9),  (2,3,⋆),  (2,3,14), (2,7,9), (3, 4, 8), (3,4,10),  (3,4,14), (3,7,8), (3,7,9), (3,7,10),  
       (3,7,14), (3,8,9), (3,8,10), (3,8,14), (3,9,10),  (4, 6,12), (4, 6,14), 
     (4, 8,10), (4, 8,14), (4, 10,12), 
         (5, 6, 13), (5, 6, 14), (6,12, 13), (7, 8, 9), (7, 8, 10), (7, 8, 14),  (7, 9, 10), (8, 9, 10)}.

Hence, 𝒞_3=𝒞^(1)_3∪𝒞^(2)_3 where 𝒞^(1)_3={{l_0, l_j, l_k}: (j, k)∈ I_2, j, k≠⋆} and
 

    𝒞^(2)_3={   {l_1, l_2, l_3},{l_1, l_2, l_⋆}, {l_1, l_2, l_9}, {l_1, l_3, l_9},  {l_1, l_9, l_10}, {l_1, l_9, l_11}, {l_3, l_4, l_8}, {l_3, l_4, l_14}, 
       {l_4, l_6, l_14}, {l_4, l_8, l_10}, {l_4, l_8, l_14},  {l_5, l_6, l_14}}.

















First consider 𝒞^(1)_3.  If (j, k)∈ I^(2)_1 we get {l_0, l_0, l_k} (1≤ k≤ 14), and the proof is similar to that of 𝒞^(1)_2. If (j, k)∈ I^(2)_2 we get {l_0, l_1, l_k} (1≤ k≤ 13, k≠ 7, 8). We compute from Table <ref> that for each k with k≠ 2, 9 we have that ϵ'_i=l_0=N'_i=M_1=min (M_1, p_1).  For k=2 we get that ϵ'_i=N_i=l_0=p_1=min(M_1, p_1). Similarly for k=9 we compute that ϵ'_i=s'_i+N'_i=min(l_1, l_9)+l_0=min(M_1, p_1) (L_i contains half twisted scissors {l_1, l_9}).  If (j, k)∈ I^(3)_2 we have ϵ'_i=l_0=M_1=min(M_1, p_1) since M_1=0 for each  corresponding (l_j, l_k); and for {l_0, l_4, l_8} we have ϵ'_i=z'_i+N'_i=min(l_4, l_8)+l_0=M_1=min(M_1, p_1) (L_i contains half twisted anchor {l_4, l_8}). Finally for  {l_j, l_k, l_m}∈𝒞^(2)_3 we similarly verify from Table <ref> that ϵ'_i=p_1=0 for (i, j, k)∈{(1, 2, 3),(1, 2, ⋆)}; ϵ'_i=s'_i=min(M_1, p_1) for (i, j, k)∈{(1, 2, 9), (1, 3, 9),  (1, 9, 10), (1, 9, 11)}; ϵ'_i=M_1=min(M_1, p_1) for  {l_3, l_4, l_8},  {l_4, l_8, l_10}, {l_4, l_8, l_14} and ϵ'_i=M_1=0=min(M_1, p_1)  
for {l_3, l_4, l_14}, {l_4, l_6, l_14}, {l_5, l_6, l_14}. Similarly,  we  have I_4=I^(1)_4∪ I^(2)_4∪ I^(3)_4 where 

    I^(1)_4={(0, j, k, m): (j, k, m)∈ I_3, j, k, m ≠⋆}, I^(2)_4={(1, j, k,  m):  (j, k, m)∈ I_3, j, k, m∉{7, 8}}
 and I^(3)_4={(2,3,7,9),  (3, 4, 8,10), (3,7,8, 9), (3,7, 8,10),( 3,8, 9,10),  (7, 8,9,10)}.

We get  𝒞_4=𝒞^(1)_4∪𝒞^(2)_4 where 𝒞^(1)_4={{l_0, l_j, l_k, l_m}:(j, k, m)∈ I_3, j, k, m ≠⋆} 


    𝒞^(2)_4={{   l_1, l_2, l_3, l_9}, {l_1, l_2, l_3, l_⋆},  {l_1, l_3, l_9, l_10},   {l_3, l_4, l_8, l_10},  {l_3, l_4, l_8, l_14}}.
  Also,  𝒞_5=𝒞^(1)_5∪𝒞^(2)_5 where 𝒞^(1)_5={{l_0, l_j, l_k, l_m, l_n}: (j, k, m, n)∈ I_4,  j, k, m, n ≠⋆} and 𝒞^(2)_5=∅ since there is no 5 element compatible set which doesn't contain l_0 but satisfies property P. And finally,  𝒞_6={l_0, l_3, l_7, l_8, l_9, l_10}. We note that there is no 𝒞_k with k≥ 6. The verification of the formula for k=4, 5, 6 is analogous. 


  Case 2.   χ_i>0: At least a component of L_i∖ I belongs to the set S_1. There are 2 subcases depending on whether or not 
L_i contains a highly twisted arc X with χ(X)>1.
 


  (a) No component  X of L_i has χ(X)>1: Then L_i contains a highly twisted arc X which has χ(X)=1 (Figure <ref>). Let 𝒜_k denote the set of arcs  which are compatible with the arc l_k∈ S_1. Then, 

    𝒜_15   ={l_0, l_8, l_9, l_10, l_12, l_16, l_19, l_20}, 𝒜_16={l_0, l_9, l_11, l_10, l_15, l_20},𝒜_17={l_0, l_8, l_9, l_18, l_19},
    𝒜_18   ={l_0, l_7, l_9, l_17, l_20},𝒜_19={l_0, l_4, l_6, l_8, l_10, l_17, l_15},𝒜_20={l_0, l_5, l_6, l_11, l_15, l_18}
 

 Any compatible set containing  l_k∈ S_1 is constructed from elements of 𝒜_k such that property P is satisfied.  Therefore,  a compatible set  can contain the standard exceptional arc l_0 (which is compatible with each element of S_1) and the twisted exceptional arcs l_19 and l_20. The only exceptional arc system  can contain is the negatively half twisted ribbon which is the arc system  {l_12, l_15}. Consider  for example the compatible sets containing l_15. Each such set  is constructed from 𝒜_15 in such a way that  it contains at least one of l_0, l_12, l_19 and l_20 so that ψ_i+1≠ 0 (the other two assumptions λ_i≠ 0 and Δ_i+1(B)≠ 0 are satisfied by each element in S_1).   We immediately check that  for each such compatible set we have p_1≥ M_1 and M_1=N'_i+r'_i where r'_i=min(l_12, l_15).  Therefore, ϵ'_i=N'_i+r'_i=min(M_1, p_1) as required.  
 
 

 
 
 
  (b)  Some component  X of L_i has χ(X)>1: Then l_0=0, s'_i=z'_i=r'_i=0 by Remark <ref> since each exceptional arc system with respect to a clover type II contains a standard arc. Since ψ_i+1≠ 0 by assumption there exists a highly twisted exceptional arc X∈ [X_2i+2; i+1] (such arcs are the only highly twisted arcs with ψ_i+1≠ 0 and satisfying property P)  each of which increases M_1 by 1. Since M_1=0 for any other arc compatible with X we get   ϵ'_i=N'_i=min(M_1, p_1) as required.   
 
  The proof of Lemma <ref> shows that there exists compatible sets satisfying property P yet containing no exceptional arcs or arc systems. Such arc systems either contain {l_1,l_2} or the arc X_2i+12i+2=l_14 together with an arc that satisfies ψ_i+1≠ 0 such as {l_5, l_14} (note that l_14 contributes to both λ_i and Δ_i+1(B)). Using parameter p_1 and subtracting X_2i+12i+2 from λ_i and Δ_i+1(B) rules out such arc systems  giving a way to compute only exceptional arcs or arc systems.

 
 
  
  
   Reflection in the horizontal diameter of the surface conjugates each
  crosscap transposition  u_i to u_i^-1. Therefore a clover of type III is the reflection of a clover of type II along the horizontal diameter, 
and the corresponding transformation of generalized Dynnikov coordinates in max-plus notation is given by [t_i; b_i]↦[1/t_i; b_i]. For example for n>0

    (t_1,…, t_g-1,b_1,…,b_n+g-2) ↦(1/t_1,…,1/t_g-1,b_1,…,b_n+g-2).
 

 
  
  By Remark <ref> we conclude that exceptional arcs  with respect to a clover of type III can be obtained by reflecting exceptional arcs with respect to a clover of type II in the horizontal diameter. Therefore, replacing u^-1_i with u_i in Notation <ref>, we obtain the exceptional parameter e̅_̅i̅=ϵ”_i+ϵ”_i+1 for a clover of type III as given in Lemma <ref>  and Lemma <ref>. 
  
 






Let L_i be positive.  Then e̅_i=ϵ”_i+ ϵ”_i+1 where 

    ϵ”_i   =min(Λ_i, ψ_i+1, Δ̅_i+1(A), p_3);
    ϵ”_i+1   =min(Λ_i+1, ψ_i, Δ̅_i(B), p_4)



 
Given a clover of type III we have
 
    2β”_n+i+C_i=max(γ_2i-1+γ_2i+2, γ”_2i-1+γ”_2i+2)+2e̅_i.





 

 §.§ Scale equalities

 
 Let  u^-1_i(γ; β)=(γ'; β') and u_i(γ; β)=(γ”; β”).  A scale of type I has leaves γ_2i+1, γ_2i, _i, β_n+i+1; diagonals γ'_2i+1 and γ_2i+2; and a scale of type II has leaves γ_2i+1, γ_2i, _i, β_n+i-1; and diagonals γ'_2i and γ_2i-1. Reflecting these two scales along the horizontal diameter we respectively obtain a scale of type III and a scale of type IV (Figure <ref>). Observe that this is natural since a scale of type III and a scale of type IV are the u_i images of a  scale of type I and a scale of type II respectively (see Remark <ref>).



 
 
 
 
 



Let X be a component of L_i.  In what follows we shall write λ(X), λ_c_k(X), ψ_k(X) (k=i, i+1) to denote the number of non–core, core and straight components of a given component X of L∩ S'_k.  





The key idea in the proof of Lemma <ref> is that it is easy to find out  which standard arcs satisfy equality  (<ref>), (<ref>), (<ref>) and (<ref>) since there are only finitely many standard arcs to check. Similarly,  we call  arcs which don't satisfy equality  (<ref>),   (<ref>),   (<ref>)  and   (<ref>)   exceptional with respect to a scale of type I, type II, type III and  type IV respectively. 
 
    γ'_2i+1+γ_2i+2=max(γ_2i+1+γ_2i, 	C_i+2β_n+i+1)

 
    γ'_2i+γ_2i-1=max(γ_2i+1+γ_2i, 	C_i+2β_n+i-1)


    γ”_2i+2+γ_2i+1=max(γ_2i+2+γ_2i-1, 	C_i+2β_n+i+1)


    γ”_2i-1+γ_2i=max(γ_2i-1+γ_2i+2, 	C_i+2β_n+i-1)



We say that X is straight in S'_k (k=i, i+1) if ψ_k(X)≠ 0. Similarly, we say that X is u_i–straight in S_k if ψ_k(u_i(X))≠ 0 for some k=i, i+1.  Definition for  u^-1_i–straight arc in S_k is similar. 

 
 Analysis of exceptional arcs with respect to a scale of type I shows that each standard component X of L_i which isn't straight  in S'_i+1 (i.e. those with ψ_i+1=0) satisfies equality (<ref>).  An analogous statement for a twisted component in the class of X is also true since each such component has the same  number of intersections with _i as X, and increases the number of intersections on each γ'_2i+1, γ_j (2i≤ j≤ 2i+2) and 2β_n+i (and hence C_i+2β_n+i+1)  by the same amount.  Also the only standard straight arcs in   S'_i+1 which satisfy equality (<ref>) are X_2i+2; i+1^(0,1;-1,0) and X_2i-1; i+1 (Figure <ref>(a)). That is, each standard straight arc in S'_i+1  apart from X_2i+2; i+1^(0,1;-1,0) and X_2i-1; i+1 is  exceptional with respect to a scale of type I (top row of Figure <ref>). Also the only twisted arcs which are straight in S'_i+1 are in the class of [X_k; i+1] (2i-1≤ k≤ 2i+2) and [X_i] (examples of which are as given on the bottom row of  Figure <ref>). Each  such exceptional arc X satisfies 
    γ'_2i+1+γ_2i+2=max(γ_2i+1+γ_2i, 	C_i+2β_n+i+1)+2ψ_i+1(X)







Since a scale of type III is the u_i image of a  scale of type I it follows from Remark <ref> that  each arc X with ψ_i+1(X)≠ 0  apart from X_2i+1; i+1^(0,1;-1,0) and X_2i; i+1 is exceptional with respect to a scale of type III. For the same reason, each u_i–straight arc X in S'_i+1 (Figure <ref>) apart from X^(1,0;-1,0)_2i+12i+2 and X_2i+2 ; i    (Figure <ref>(b)) is exceptional with respect to a scale of type I. Let us write ψ_k(u_i(X))=ψ_k(X') and  ψ_k(u^-1_i(X))=ψ_k(X”). Then for each u_i–straight arc X in S'_i+1 we get
 
    γ'_2i+1+γ_2i+2=max(γ_2i+1+γ_2i, 	C_i+2β_n+i+1)+2ψ_i+1(X')

  Write  ψ_k(u_i(L))=ψ'_k and  ψ_k(u^-1_i(L))=ψ”_k for a given L∈ℒ_n. 

Then, setting 
    f_i=ψ_i+1-(X_2i+2; i+1^(0,1;-1,0)+X_2i-1; i+1)+ψ'_i+1-(X^(1,0;-1,0)_2i+12i+2+X_2i+2 ; i)

we obtain equality (<ref>) given in Lemma <ref>. We call f_i the exceptional parameter for a scale of type I. The exceptional parameters for a scale of type II, type III and type IV follow from symmetry:

 


 
    g_i   =ψ_i-(X_2i-1; i^(1,0;0,1)+X_2i+2; i)+ψ'_i-(X^(0,1;0,1)_2i-12i+X_2i-1; i+1)

 
    f̅_̅i̅   =ψ_i+1-(X_2i+1; i+1^(0,1;-1,0)+X_2i; i+1)+ψ”_i+1-(X^(1,0;-1,0)_2i+12i+2+X_2i+1 ; i)

  
    g̅_̅i̅   =ψ_i-(X_2i; i^(1,0;0,1)+X_2i+1; i)+ψ”_i-(X^(0,1;0,1)_2i-12i+X_2i; i+1)


Hence, computing f_i, f̅_̅i̅, g_i, g̅_̅i̅ in terms of generalized  Dynnikov coordinates will require separate consideration of the arcs depicted in Figure <ref> and given in Lemma <ref> and Lemma <ref>. We first state scale equalities in Lemma <ref>.




 
 
  
Given a scale of type I, type II, type III and type IV as shown in Figure <ref> we have 
 
    γ'_2i+1+γ_2i+2=max(γ_2i+1+γ_2i, 	C_i+2β_n+i+1)+2f_i

 
    γ'_2i+γ_2i-1=max(γ_2i+1+γ_2i, 	C_i+2β_n+i-1)+2g_i


    γ”_2i+2+γ_2i+1=max(γ_2i+2+γ_2i-1, 	C_i+2β_n+i+1)+2f̅_̅i̅


    γ”_2i-1+γ_2i=max(γ_2i-1+γ_2i+2, 	C_i+2β_n+i-1)+2g̅_̅i̅

 
 



 Consider the arcs X_k; j (2i-1≤ k≤ 2i+2, j=i,i+1) in Figure <ref>, and let χ(i)<1. Then,

    X_2i-1; i+1   =min(ψ_i+1, [χ_i(A)]^+ )   X_2i; i+1=min(ψ_i+1, [χ_i(B)]^+ )
    X_2i+1;i    =min(ψ_i, [χ_i+1(A)]^+ )    X_2i+2; i=min(ψ_i, [χ_i+1(B)]^+ )







We prove X_2i-1; i+1=min(ψ_i+1, [χ_i(A)]^+ ). The other equalities can be proved in a symmetric way. The proof is similar to the proof of Lemma <ref>, and is based on the following facts:


  (1) X_2i-1; i+1 increases ψ_i+1 and χ_i(A) by 1. 

  (2) If X is compatible with X_2i-1; i+1 then χ_i(A)<1.

Therefore, by fact (1) if ψ_i+1=0 or χ_i(A)=0, then X_2i-1; i+1=0. Similarly, by fact (2) if χ(i)>1 then X_2i-1; i+1=0. So suppose that  ψ_i+1≠ 0 and χ_i(A)≠ 0, and that χ(i)<1 which is guaranteed by the assumption of the lemma.  Let us say that an arc X has property Q if it satisfies ψ_i+1≠ 0,  χ_i(A)≠ 0 and χ(X)<1; and that an arc is compatible with property Q if it is compatible with an arc that satisfies property Q. Figure <ref> illustrates all arcs apart from X_2i-1; i+1 which are compatible with property Q. Since [χ_i(A)]^+=0 for each of these arcs X_2i-1; i+1=min(ψ_i+1, [χ_i(A)]^+ ) by fact (1).


 









 
Consider the arcs  X^v_i_k; j (2i-1≤ k≤ 2i+2, j=i,i+1)  in Figure <ref>.  Let

    M_1   =min(Λ_i, ψ_i+1-X_2i-1; i+1, Δ̅_i+1(B)); M_2=min(Λ_i+1, ψ_i-X_2i+2; i, Δ̅_i(A))
    
    M_3   =min(Λ_i, ψ_i+1-X_2i-1; i+1, Δ̅_i+1(B)); M_4=min(Λ_i+1, ψ_i-X_2i+2; i, Δ̅_i(A))

Then


    X_2i+2; i+1^(0,1;-1,0)   =  min(M_1, (c_i+1-(c_i-|χ_i+1(B)|))^+, (c_i+1-2[χ_i+1(B)]^+)^+)
    
     X_2i+1; i+1^(0,1;-1,0)   =min(M_4, (c_i+1-(c_i-|(χ_i+1(A)|))^+, (c_i+1-2[χ_i+1(A)]^+)^+)
    
    X_2i-1; i^(1,0;0,1)   = min(M_2, (c_i-(c_i+1-|χ_i(A)|))^+, (c_i-2[χ_i(A)]^+)^+) 
    
    X_2i; i^(1,0;0,1)   =min(M_3, (c_i-(c_i+1-|χ_i(B)|))^+, (c_i-2[χ_i(B)]^+)^+)





We compute X_2i+2; i+1^(0,1;-1,0) which is a standard exceptional arc with respect to a clover of type II. Again, the other equalities can be proved in a symmetric way.  To compute this arc separately  we need modification on the formulae given in Lemma <ref> to eliminate the values s'_i, r'_i, z'_i which are parameters related with exceptional arc systems of type II, and the number of highly twisted exceptional arcs in the set [X_2i+2; i+1]. Using  the value X_2i-1; i+1- ψ_i+1 in M_1 rules out the possibility of scissors and hence guarantees that s'_i=0. Similarly, since (c_i+1-(c_i-|χ_i+1(B)|))^+=0 for  anchors and ribbons we get r'_i=z'_i=0. Finally, for each highly twisted exceptional arc in [X_2i+2; i+1^(0,1;-1,0)] we have  min( (c_i+1-(c_i-|χ_i+1(B)|))^+, (c_i+1-2[χ_i+1(B)]^+)^+)=0 (see for instance l_19 and l_20).  Since X_2i+2; i+1^(0,1;-1,0) increases M_1, (c_i+1-(c_i-|χ_i+1(B)|))^+ and (c_i+1-2[χ_i+1(B)]^+)^+ by one we conclude that X_2i+2; i+1^(0,1;-1,0) is as given in equation (<ref>).











 
 
 





Let ψ_k(u_i(L))=ψ'_k and  ψ_k(u^-1_i(L))=ψ”_k (k=i, i+1) denote the number of straight components of u_i(L)∩ S'_k and u^-1_i(L)∩ S'_k respectively. 

Let L_i be negative. Then

    ψ'_i+1= λ^-_c_i+χ_i+1(B)     if χ_i+1(B))≤ 0
    
                 (Δ_i+1(B)-λ_i^-)^+   if χ_i+1(B))>0
             and  ψ'_i= λ^+_c_i+1+χ_i(A)   if χ_i(A)≤ 0
    
                 (Δ_i(A)-λ^+_i+1)^+   if χ_i(A)>0

Let L_i be positive. Then,

    ψ”_i+1= λ^-_c_i+χ_i+1(A)    if χ_i+1(A))≤ 0
                 (Δ_i+1(A)-λ_i^-)^+   if χ_i+1(A))>0
             and  ψ”_i= λ^+_c_i+1+χ_i(B)   if χ_i(B)≤ 0
    
                 (Δ_i(B)-λ^+_i+1)^+   if χ_i(B)>0




To compute the number ψ'_i+1 of straight components  of u_i(L)∩ S'_i+1 we need to determine which arcs are u_i-straight in S'_i+1. In order to do this, we first list all standard arcs which are straight  in S'_i+1 (there are finitely many of those) and take their inverse images under u_i from which we obtain the arcs depicted in Figure <ref>. Using Notation <ref>  we write the following facts:

 



  (1) Each u_i-straight arc X is negative with χ_i+1(B)=0 and λ^-_c_i(X)≠ 0 (i.e. X has a left core-loop in S'_i). The converse is also true.

  (2) ψ_i+1(u_i(X)) equals the number of left core loops 
of X∩ S'_i which are entirely contained in below components of X∩ S'_i+1.

We have the following cases:




  * If  χ_i+1(B)<0, L_i contains X^(1, 0; -1, 0)_2i+22i+2  which satisfies λ^-_c_i≠ 0, is not u_i-straight and not compatible with any highly twisted component. Therefore, by (1) and (2) we obtain 
ψ'_i+1=λ^-_c_i-X^(1, 0; -1, 0)_2i+22i+2. It is easy to show that 
X^(1, 0; -1, 0)_2i+22i+2=-χ_i+1(B) from which we get  
ψ'_i+1=λ^-_c_i+χ_i+1(B). Clearly, if χ_i+1(B)=0, L_i is some collection of arcs depicted in Figure  <ref> each of which satisfies
 ψ'_i+1=λ^-_c_i=(Δ_i+1(B)-λ_i^-)^+ by (2).
 
 
  * If χ_i+1(B)>0, L_i contains a highly twisted component X, and only left core loops of X∩ S_i that do not join right loop components of X∩ S_i+1 (i.e. those that are contained in  Δ_i+1(B)) can be mapped to a straight component of X∩ S_i+1. That is for each such arc we have ψ'_i+1=(Δ_i+1(B)-λ_i^-)^+. 
 
 
 
 












Let n>0 and 𝒜_g,n denote the set of arcs in Figure <ref>.   (N_g,n) acts on both 𝒜_g,n and 𝔏_g,n,  and hence i(, ξ)=i(δ(),δ(ξ)) for any δ∈(N_g,n) and ξ∈_g,n.  We also recall that  the arcs α_i (1≤ i≤ 2n-2) and β_i (1≤ i≤ n) are not affected by crosscap transpositions. For the crosscap transposition u_i, our approach is to compute the number of intersections of γ'_j=u^-1_i(γ_j) (1≤ j≤ 2g-2) and β'_j=u^-1_i(β_j) (1≤ j≤ n+g-1)  with  instead of computing the number of intersections of u_i() with γ_j and β_j.  We have,

    t'_j=γ'_2j-γ'_2j-1/2  and   b'_j=β'_j-β'_j+1/2.

We shall make use of clover and scale equalities given in Lemma <ref>, Lemma <ref>, Lemma <ref>.   For computational convenience we set T_j=2t_j (1≤ j≤ g+n-2), B_j=2b_j (n≤ j≤ g+n-2), 2β_n+j=𝔅_n+j (1≤ j≤ g-1) and D_j=2d_j, E_j=2e_j, F_j=2f_j, G_j=2g_j  (1≤ j≤ g-1), and work in the max-plus semiring as indicated in Remark <ref>.  Therefore, 


    T_j=[γ_2j/γ_2j-1] and B_j=[β_j/β_j+1].

and from the clover of type I equality  (<ref>), 


    _i=[D_i(γ_2i-1γ_2i+2+γ_2i+1γ_2i)/𝔅_n+i]





We now consider the two separate cases of the statement.



  * Suppose that  1≤ i < g+n-2/ Observe that β'_j=β_j for j≠ n+i and γ'_j=γ_j for j<2i-1 and j>2i+2. Therefore, T'_j=T_j for except j=i and j=i+1; and B'_j=B_j for except j=n+i-1 and j=n+i. Next we compute T'_i, T'_i+1, B'_n+i-1 and  B'_n+i.





  * 
We shall first compute T'_i+1=[γ'_2i+2/γ'_2i+1]. We have γ'_2i+2=γ_2i. To compute γ'_2i+1 we use the scale of type I equality (<ref>) and obtain

    γ'_2i+1= [F_i(_i𝔅_n+i+1+γ_2iγ_2i+1)/γ_2i+2].




 
Then from (<ref>), (<ref>) and (<ref>) we compute that


    [1/T'_i+1]   =   [F_i(D_i(γ_2i-1γ_2i+2+γ_2i+1γ_2i)/𝔅_n+iγ_2iγ_2i+2𝔅_n+i+1+γ_2iγ_2i+1/γ_2iγ_2i+2)]
    
    and hence,
    
    T'_i+1   =   [T_iT_i+1B^2_n+i/F_i(T_i(D_i+B^2_n+i)+D_iT_i+1)]. That is,
    
    T'_i+1   =   T_i+T_i+1+2B_n+i-(F_i+max(T_i+max(D_i, 2B_n+i), D_i+T_i+1))

Dividing both sides of the equation by 2 we get  
    t'_i+1=[t_it_i+1B_n+i/f_i(t_i(d_i+B_n+i)+ d_it_i+1)]=[t_it_i+1b^2_n+i/f_i(t_i(d_i+b^2_n+i)+ d_it_i+1)].



  * 
We shall now compute T'_i=[γ'_2i/γ'_2i-1]. We have γ'_2i-1=γ_2i+1. To compute γ'_2i  we use the scale of type II equality (<ref>) and obtain

    γ'_2i=[G_i(γ_2iγ_2i+1+C_i𝔅_𝔫+𝔦-1)/γ_2i-1].


Hence, from (<ref>) we get 
    t'_i=[g_i(t_i(1+d_ib^2_n+i-1)+d_ib^2_n+i-1t_i+1)].



  *  We proceed with B'_n+i=[β'_n+i/β'_n+i+1]. We have β'_n+i+1=β_n+i+1 and from the clover of type II equality (<ref>), 





    𝔅'_n+i=[E_i(γ'_2iγ'_2i+1+γ_2iγ_2i+1)/C_i].

 Since γ'_2i-1=γ_2i+1 and γ'_2i+2=γ_2i  
    γ'_2i=[T'_iγ_2i+1]  and  γ'_2i+1=[γ_2i/T'_i+1].












    B'_n+i   =[E_i(T'_i/T'_i+1γ_2iγ_2i+1+γ_2iγ_2i+1)/C_i𝔅_n+i+1]=[E_iγ_2iγ_2i+1(T'_i/T'_i+1+1)/D_i𝔅_n+i+1/𝔅_n+i(γ_2i-1γ_2i+2+γ_2iγ_2i+1)]
       =[E_i/D_iB^2_n+i(T'_i+T'_i+1/T'_i+1)1/γ_2i-1γ_2i+2+γ_2iγ_2i+1/γ_2iγ_2i+1]=[E_i/D_iB^2_n+i(T'_i+T'_i+1/T'_i+1)T_i/T_i+T_i+1]

from which we get 
    b'_n+i=[e_i/d_ib^2_n+i(t'_i+t'_i+1/t'_i+1)t_i/t_i+t_i+1].



 
  * Now we shall  compute B'_n+i-1=[β'_n+i-1/β'_n+i]. We have
 
    β'_n+i-1=β_n+i-1, β'_n+i+1=β_n+i+1 and B'_n+i=[β'_n+i/β'_n+i+1].


 Therefore, B'_n+i-1=[β_n+i-1/B'_n+iβ_n+i+1]. Multiplying the numerator and denominator by β_n+i gives B'_n+i-1=[B_n+iB_n+i-1/B'_n+i]. That is,  
    b'_n+i-1=[b_n+ib_n+i-1/b'_n+i].

 
 
 


  * Now, suppose that i=g-1 (Figure <ref>). Observe as before that t'_j=t_j for all j<g-1 and b'_j=b_j for all j<n+g-2. Since there are no teardrops encircling the last crosscap, our approach to compute t'_g-1 and b'_n+g-2 is to add  dummy teardrops γ_2g-1, γ_2g and β_n+g as depicted in Figure <ref>, which enables us to make similar calculations as in the previous statement. We first note that γ_2g-1=γ_2g=β_n+g-1/2 and β_n+g=0 hence we have T_g=0 and B_n+g-1=β_n+g-1. Similar calculations give


    t_g-1' = g_g-1(t_g-1+d_g-1B_n+g-2(1+t_g-1))  and   B_n+g-2' = d_g-1/e_g-1B_n+g-21+t_g-1/t_g-1(1+t'_g-1)






 

  

Now let n=0. The formulae for 1<i≤ g-2 are obtained similarly replacing i with i-1. For i=1, we add two dummy punctures around the first crosscap. Similar arguments  give that


  
    t_1' =[t_1B_1/f_0(1+B_1)+ t_1]   and   B_1' = [B_1e_0(1+t'_1)/t_1'1/(1+t_1)]

  
For i=g-1 we note that rotation through π about the center of the surface conjugates each
crosscap generator u_i to u_g-i and the corresponding
  transformation of generalized Dynnikov coordinates in max-plus notation, is given by

    (t_1,…, t_g-2,b_1,…,b_g-2) ↦(1/t_g-2,…, 1/t_1,1/b_g-2,…, 1/b_1)

hence we get

    t_g-2' = g_g-2(t_g-2+d_g-2B_g-2(1+t_g-2))  and   B_g-2' = d_g-2/e_g-2B_g-21+t_g-2/t_g-2(1+t'_g-2)





By Remark <ref> we obtain the rules for t”_i and b”_i for each case by symmetry, conjugating the rules for u_i by the involution (<ref>).































 Note that the method introduced in this paper can be used to provide an efficient way to solve on non–orientable surfaces many combinatorial and dynamical problems <cit.>  that were previously solved only on orientable surfaces before  <cit.>. 
However, to solve such problems not only for sequences of crosscap transpositions but any element of the mapping class group we need to  describe the action of the mapping class group (N_g,n)  on 𝔏_g,n   in terms of generalized Dynnikov coordinates <cit.>. That is we need to  compute the action of the other generators of (N_g,n) which are crosscap slides, puncture slides and Dehn twists about certain 2-sided curves <cit.> in terms of generalized Dynnikov coordinates <cit.>    which require similar techniques introduced in this paper.





This work was completed during a visit of the author at  Columbia University as a Fulbright scholar. The author would like to thank the Fulbright Scholar Program for their support and  Columbia University for their warm hospitality. 




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