Coherent double coverings of virtual link diagrams
Naoko Kamada
math.GT



Coherent double coverings of  virtual link diagrams]
Coherent double coverings of  virtual link diagrams
 
This work was supported by JSPS KAKENHI Grant Number 15K04879.

 Graduate School of Natural Sciences,  Nagoya City University
 
1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501 Japan






 
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. 
Normal virtual links have some properties similar to classical links.



In  this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. 
We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent.
 We discuss the relationship between this 
method and some invariants of virtual links.


[
    Naoko Kamada
    
================










§ INTRODUCTION





Virtual links correspond to  stable equivalence classes of  links in thickened surfaces <cit.>. 

A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable ( <ref>). A virtual link is called normal or checkerboard colorable if it has a normal diagram as a representative. Every classical link diagram is normal, and hence 
the set of classical link diagrams is a subset of that of normal virtual link diagrams. The set of normal virtual link diagrams is a subset of that of virtual link diagrams. 
Jones polynomial is extend to virtual links <cit.>. It is shown in <cit.> that Jones polynomial of a normal virtual link has a property that Jones polynomial of a classical link has. 
For this property Khovanov homology is extended to normal virtual links in a natural way as stated in O. Viro <cit.>.
The author introduced the method of  converting a virtual link diagram to a normal virtual link diagram by use of  the double covering technique in <cit.>. In this method, two converting virtual link diagrams obtained from two equivalent virtual link diagrams are related by generalized Riedemeister moves and K-flypes (See <cit.>). 

In  this paper, we introduce another method of converting a virtual link diagram to a normal virtual link diagram, which is called the coherent double covering technique. 
We show that two normal virtual link diagrams obtained from two equivalent virtual link diagrams by our method 
are equivalent as a virtual link.  In Section <ref>, we discuss the relationship between our method and some invariants of virtual links, as the odd writhe, the invariant of even ordered virtual link which is introduced by H. Miyazawa, K. Wada and Y. Yasuhara<cit.>.




§ DEFINITIONS AND MAIN RESULTS
 
A virtual link diagram is a generically immersed, closed and oriented 1-manifold in ℝ^2 with information of positive, negative or virtual crossing, on each double point.  A virtual crossing is an encircled double point without over-under information <cit.>. 











A virtual link  is an equivalence class of virtual link diagrams under Reidemeister moves and virtual Reidemeister moves depicted in Figure  <ref>. We call Reidemeister moves and virtual Reidemeister moves  generalized Reidemeister moves. 




An abstract link diagram (ALD)  is a pair (Σ, D) 
of  a compact surface Σ and a link diagram D on Σ such that the underlying 4-valent graph 
|D| is a deformation retract of Σ. 










We obtain an ALD from a virtual link diagram D by corresponding a diagram to an ALD as in Figure <ref> (i) and (ii).
Such an  ALD is called the ALD associated with D.
Figure <ref> shows a virtual link diagram and the ALD associated with it. 
For details on abstract link diagrams and their relations to virtual links, refer to <cit.>.





Let D be a virtual link diagram and (Σ, D_Σ) the ALD associated with D. 
The diagram D is said to be normal or checkerboard colorable if the regions of Σ-|D_Σ| can be colored black and white such that colors of two adjacent regions are different. In Figure <ref>, we show an example of a normal diagram. (The orientations of the diagrams in Figure <ref> are alternate orientations, which are discussed later.) A classical link diagram is normal. A virtual link is said to be normal if it has a normal virtual link diagram. Note that normality is not necessary to be 
preserved  under generalized Reidemeister moves. 
For example the virtual link diagram in the right of Figure <ref> is not normal and is equivalent to the trefoil knot diagram in the left which is normal. 






H. Dye introduced the notion of cut points on a virtual link diagram in <cit.>.



Let (D, P) be a pair of a virtual link diagram D and a finite set P of points on edges of D. 
We obtain an  ALD from (D, P) as in Figure <ref> (i), (ii) and (iii) and  call it the ALD associated with (D, P).
 See Figure <ref> (ii) and (iii). 
If the ALD associated with (D, P) is normal,  then we call the set of points P  a cut system of D and call each point of P a cut point.

For  the virtual link diagram  in Figure <ref> (i) we show an example of a cut system  in Figure <ref> (ii)  and the  ALD associated with it with a checkerboard coloring in Figure <ref> (iii). 
















A virtual link  diagram is said to admit an alternate orientation if it can be given an orientation  such that an  orientation of an edge switches at each classical crossing  as in Figure <ref>. 
(Here an edge means an edge of |D|, where endpoints are classical crossings and there might be some virtual crossings on it.) 
The virtual link diagram in Figure <ref> admits an alternate orientation. 
It is known that a virtual link diagram is normal if and only if it admits an alternate orientation <cit.>. 














Note that a finite set P of points on D is a cut system if and only if (D,P) admits an alternate orientation such that the orientations  of edges are as in Figure <ref> at each classical crossing of D and each point of P (cf. <cit.>).

The canonical cut system of a virtual link diagram D  is the set of points 
 that is obtained by 
giving two points in a neighborhood of each virtual crossing of D as in Figure <ref> (i). 




The canonical cut system is a cut system.








Dye introduced cut point moves  depicted in Figure <ref>.
Then we have the following.







For a virtual link diagram D, two cut systems of  D are related by a sequence of cut point moves I, II and III.
















For any virtual link diagram with a cut system,  the number of cut points is 
even.





Let D be a virtual link diagram with a cut system P. 
Assume that (D, P) is on the right of the y-axis in the xy-plane and all crossings and cut points have distinct y-coordinates.  Let  (D^*, P^*) be a copy of (D,P) on the left of the y-axis in the xy-plane which is obtained from (D,P) by sliding along the x-axis. Let {p_1,…, p_k} be the set of cut points of P and for i∈{1,…, k}, we denote  by p_i^* the cut point of D^* corresponding to p_i.

See Figure <ref> (i). 



For horizontal lines l _1, … , l_k such that  l_i contains p_i and the corresponding cut point p_i^* of D^*, 
we  replace  each part of D⨿ D^* in a neighborhood of N(l_i) for each i∈{1,…, k}  as in Figure <ref>. We denote by ϕ(D, P) the virtual link diagram obtained this way.

For example, for the virtual link diagram D with the cut system P  depicted as in Figure <ref> (i), the virtual link diagram ϕ(D, P) is as in Figure <ref> (ii).









Then we have the following.









For a virtual link diagram D with a cut system P, (D, P),   ϕ(D, P) is normal.


Let D be a virtual  link diagram D with a cut system P. The virtual link diagram D^* with the cut system P^* is a copy of  (D, P) as the previous manner as in Figure <ref> (i). The  ALDs obtained from D and D^* can be colored as in Figure <ref> (ii). Then we see that ϕ(D, P) is normal as in the right of Figure <ref> (ii). 






































Then we have 

    ϕ:{}⟶{}.




Precisely speaking, ϕ(D,P) is well-defined when (D,P) satisfies that all crossings and cut points have distinct y-coordinates. If we change (D,P) by an isotopy of ℝ^2, ϕ(D,P) is preserved under an isotopy of ℝ^2 and virtual Reidemeister moves.


Let  (D,P) be a virtual link diagram with a cut system P. We call ϕ(D,P) the converted normal diagram or the coherent double covering diagram of (D, P). 

Let D be a virtual link diagram. For the canonical cut system P_0 of D, we denote by ϕ_0(D) the converted normal diagram ϕ(D, P_0). 
We call ϕ_0(D) the canonical converted normal diagram of D.
Then we have 

    ϕ_0:{}⟶{}.

















The following is our main theorem.


Let (D,P) and (D',P') be virtual link diagrams with cut systems. If D and D' are equivalent, 
then the converted normal diagrams ϕ(D,P) and ϕ(D',P') are equivalent. 
In particular, if D and D' are equivalent, 
then the canonical converted normal diagrams ϕ_0(D) and ϕ_0(D') are equivalent.









§ PROOF OF THEOREM <REF>

Theorem <ref> is obtained from Lemmas <ref> and <ref> stated below. 

Let D be a virtual  link diagram. Suppose that P and  P' are cut systems of D.
Then the converted normal diagrams ϕ(D,P) and ϕ(D,P') are equivalent.


Let D be a virtual  link diagram with a cut system P. Suppose that  P' is a cut system of D obtained from P by one of cut point moves I , II or III in Figure <ref>. 
Then ϕ(D, P') is related to ϕ(D, P) by  virtual Reidemeister moves II and III as in Figure <ref>. Thus ϕ(D, P) and ϕ(D, P') are equivalent. 







Let D_1 and D_2 be virtual link diagrams.
If D_1 and D_2 are  equivalent then the canonical converted normal virtual link diagrams ϕ_0(D_1) and ϕ_0(D_2) are equivalent.


Let D_1 be a virtual  link diagram with the canonical cut system P_1. Suppose that a virtual link diagram D_2 is obtained from D_1 by one of generalized Reidemeister moves   
and P_2 is the canonical cut system of D_2.
If D_2 is related to D_1 by one of Reidemeister moves, then ϕ_0(D_1)=ϕ(D_1, P_1) and ϕ_0(D_2)=ϕ(D_2,P_2) are related by two Reidemeister moves. 


As in Figure <ref> (i) (or (ii)), suppose that  D_2 is related to D_1  by a virtual Reidemeister move I (or II) and let P_2' be the cut system obtained from P_2 by cut point moves I and II as in the figure. 
By Lemme <ref>,  ϕ(D_2,P_2) and ϕ(D_2,P_2') are equivalent.
On the other hand ϕ(D_1,P_1) and ϕ(D_2,P_2') are related by two virtual Reidemeidter moves I (or II) as in Figure <ref> (i) (or (ii)). Thus ϕ(D_1,D_1) and  ϕ(D_2,D_2) are equivalent.
As in Figure <ref> (iii), suppose that  D_2 is related to D_1  by a virtual Reidemeister move III and let P_1' (or P_2') be the cut system obtained from P_1 (or P_2) by cut point moves I and II as in the figure. 
By Lemme <ref>,  ϕ(D_1,P_1)  (or ϕ(D_2,P_2) ) and ϕ(D_1,P_1') (or ϕ(D_2,P_2')) are equivalent.
On the other hand, ϕ(D_1,P_1') and ϕ(D_2,P_2') are  related by two virtual Reidemeidter move III. 
As in Figure <ref>  (iv), suppose that  D_2 is related to D_1  by a virtual Reidemeister move IV and let P_1' (or P_2') be the cut system obtained from P_1 (or P_2) by cut point moves I and II  
(or cut point moves I , II and III) as in the figure. By Lemme <ref>,  ϕ(D_1,P_1)  (or ϕ(D_2,P_2) ) and ϕ(D_1,P_1') (or ϕ(D_2,P_2')) are equivalent.
On the other hand, 
as in Figure <ref>  (v),  ϕ(D_1',P_1')  and ϕ(D_2',P_2') are equivalent by virtual Reidemeister moves. 

In Figure <ref>, if the orientation of some strings of virtual link diagram D_i are different from those of it,  we have the result by a similar argument. 
 










§ A RELATIONSHIP BETWEEN INVARIANTS AND COHERENT DOUBLE COVERING DIAGRAMS


In this section we discuss a relationship between some invariants and coherent covering diagrams.



 §.§ Odd writhe

We give  
an interpretation of the odd writhe of a virtual knot in terms of the linking number of the converted normal diagram ϕ(D, P). The argument of this subsection is  similar to that of Section 4 in <cit.>. 

For a 2-component virtual link diagram  D, the half of the sum of signs of nonself classical crossings of D is called the linking number of D.  
The linking number is an invariant of 2-component virtual link.

We have the following theorems (Theorems <ref> and <ref>).


Let (D,P) be a virtual knot diagram with a cut system. 
Then ϕ(D,P) is a 2-component virtual link diagram and the linking number of ϕ(D,P) is an invariant of the virtual knot represented by D.


The odd writhe is a numerical invariant of virtual knots <cit.>. We recall the definition of the odd writhe later. 


Let (D,P) be a virtual knot diagram with a cut system. The linking number of ϕ(D,P) is equal to the odd writhe of D.




Let D be a virtual link diagram. The Gauss diagram of D is a set of oriented circles which are the preimage of D with oriented chords each of which corresponds to a classical crossing and  its tail (or its head) indicates an overpass  (or an underpass) of the  classical crossing. Each chord is equipped with a sign of the corresponding classical crossing. 
The Gauss diagram in Figure <ref> (i)  is that of the virtual knot diagram in Figure <ref> (i). 



For  a virtual link diagram D with a cut system P, the Gauss diagram with points, denoted by G(D,P), of (D,P) is obtained from the Gauss diagram of D by adding points on arcs which correspond to the points of P. We denote by G(D) the Gauss diagram of D and by G(P) the points of G(D,P) corresponding to P. Then G(D, P)=(G(D), G(P)). In Figure <ref> (ii) we show  the Gauss diagram with  points of the virtual link diagram with points in Figure <ref> (ii).
Let (D^*,P^*) be the virtual link diagram with a cut system which is a copy of  (D,P).
The Gauss diagram with points G(D^*,P^*) is a copy of  G(D,P).  

The Gauss diagram of ϕ(D,P) is obtained from the Gauss diagrams of D⨿ D^* by a local replacement around each point p∈ G(P) of the Gauss diagram G(D) and around the corresponding point p^*∈ G(P^*) of G(D^*) as in  Figure <ref>.  For a virtual knot diagram of D with a set of points on edges P, we denote the Gauss diagram of ϕ(D,P) by G(ϕ(D, P)). 
The Gauss diagram in Figure <ref> (iii) is G(ϕ(D,P)) for (D, P) depicted in Figure <ref> (ii) by the map ϕ. 









For the Gauss diagram of (D⨿ D^*, P⨿ P^*), suppose that p_1, …, p_n are points in P such that the point p_i+1 follows the point p_i along the orientation of D and the point p_i^* in P^*  corresponds to the point p_i. 
In what follows, we denote by the same symbol p_i (or p_i^*) for the point p_i of P (or p_i^* of P^*) and the corresponding point of G(P) (or G(P^*)). 

Let A_i (or A_i^*) be the arc of the Gauss diagram of G(D⨿  D^*, P⨿ P^*) between two points p_i and p_i+1 (or p_i^* and p_i+1^*), and  the arc between two points p_n and p_1 (or p_n^* and p_1^*) is A_n (or A_n^*). Note that A_i corresponds to A_i^*. We also denote an arc of G(ϕ(D,P)) which corresponds to  A_i or A_i^*  of G(D⨿ D^*, P⨿ P^*) by A_i or A_i^*, respectively. Here A_i (or A_i^*) is the arc in G(ϕ(D,P)) which is obtained from A_i (or A_i^*) by removing a regular neighborhood of p_i and p_i+1(or  p_i^* and p_i+1^*).




Let (D,P) be a virtual knot diagram with  points P. Suppose P={p_1,…, p_2n} for a positive integer n. Then  ϕ(D,P) is a 2-component virtual link diagram D_1∪ D_2. Furthermore if an arc A_i is in G(ϕ(D, P))|_D_1 (or G(ϕ(D, P))|_D_2), then A_i+1 and A_i^* are in G(ϕ(D, P))|_D_2 (or G(ϕ(D, P))|_D_1).



We use the induction on n. 
Suppose that n=1, i.e. D is a virtual knot diagram with 2 points p_1 and p_2. The Gauss diagram G(ϕ(D, P)) is depicted as in Figure <ref>, where the bold line and the thin line indicate the different components and we dropped all chords in the figure. In this case ϕ(D, P) is a 2-component virtual link diagram. Two arcs A_1  and A_2^* are in one component of G(ϕ(D, P)), and A_2 and A_1^* are in the other.



Suppose that the statement is hold if the number of points is less than 2n.
We assume that D is a virtual knot diagram with 2n points p_1,…, p_2n. 
First, apply the replacement as in Figure <ref> to 2n-2 points p_1,…, p_2n-2 and 
p_1^*,…, p_2n-2^*. Then we obtain a Gauss diagram G with 4 points p_2n-1, p_2n and  p_2n-1^*, p_2n^*. 
By the hypothesis, the Gauss diagram G  is  depicted as in Figure <ref> (i), where two arcs A_2n-1 and A_2n (or two points p_2n-1 and p_2n) are in one component of G and two arcs A_2n-1^* and A_2n^* (or two points p_2n-1^* and p_2n^*) are in  the other. If an arc A_i is in  one component of G,  A_i^* (or A_i+1) is in the other for  i 2n-1 by the induction hypothesis.  
By applying the replacement in Figure <ref> to two pairs of points p_2n-1 and p_2n-1^* and p_2n and p_2n^* of  the Gauss diagram G, we have a Gauss diagram as in Figure <ref> (ii). Therefore we have the result.













Let (D,P) be a virtual knot diagram with a cut system. By Lemma <ref>, ϕ(D,P) is a 2-component virtual link diagram. Let (D', P') be a virtual knot diagram with a cut system such that D and D' present the same virtual knot. 
By Theorem <ref>, ϕ(D, P) and ϕ(D', P') are equivalent. Hence they have the same linking numbers.


For a virtual knot K and its diagram D with the cut system P, we denote  the linking number of ϕ(D,P) by lk_N(K) or lk_N(D). 
Theorem <ref> states that lk_N(D) is equal to the odd writhe of D.

Let D be a virtual knot diagram and G be a Gauss diagram of D. 
For a classical crossing c, we denote by γ_c the chord of G corresponding to c. The endpoints of γ_c divides the circle of G into 2 arcs. We denote  the arcs by I_c and I'_c where I_c is the arc which starts from the tail of γ_c and terminates at the head. 
A classical crossing  c of D is said to be   odd if there are an odd number of endpoints of chords of G on I_c. The odd writhe of D is the sum of signs of odd crossings of D．Note that if a virtual knot diagram is normal, all classical crossings are not odd and hence the odd writhe is zero.


It is shown in <cit.> that 
the odd writhe is an invariant of virtual knots.



Let D be a virtual knot diagram and P be a cut system of D.  
It is sufficient to show that 
odd crossings of D  correspond  to nonself classical crossings  of ϕ(D,P). 
Since (D,P) admits an alternate orientation, the circle of the Gauss diagram G(D,P)  of (D,P) admits an alternate orientation such that  one endpoint of each chord is 
a sink of the orientations and the other is a source. This implies the following condition: 


  (∗) 
For any classical crossing c of (D, P),
 the sum of the number of cut points on the arc I_c and that of endpoints of chords appearing on I_c, is even. 
 

Let A_1, A_2, … be the arcs obtained by cutting the circle of G(D,P) along the cut points.  We assume that 
 the arc A_i+1 appears after the arc A_i along the orientation of D. 
We also denote   by A_i the arc of G(ϕ(D,P))|_D which corresponds to A_i. 
For a classical  crossing c of (D, P), the classical crossing corresponding to c in ϕ(D, P)|_D is denoted by c̃. 
Let c be  an odd crossing of (D, P). 
Suppose that  one endpoint of  γ_c  
is on A_k and the other endpoint  is on A_j. By definition, there are an odd number of endpoints of chords on I_c in  G(D,P). 
By the condition (∗) above, we see that there are an odd number of cut points on I_c in  G(D,P).  
Then we see that k is not congruent to  j modulo 2.  By Lemma <ref> A_k and A_j in  G(ϕ(D,P))|_D are in distinct components of G(ϕ(D,P)). Thus the classical crossing c̃ is a nonself classical crossing of ϕ(D,P). By a similar argument, we see that if c is not an odd crossing of (D,P), then c̃ is a self classical crossing of ϕ(D,P).


We show some properties of lk_N(D). They are also obtained from the properties of the odd writhe.


Let K be a normal virtual knot. Then lk_N(K) is zero.


Let D_N be a normal knot diagram of K. 
The empty set ∅ is a cut system.
The virtual link diagram ϕ(D_N,∅) is the disjoint union, D_N ⨿ D_N^* of D_N and D^*_N. Thus lk_N(K) is zero.

Let D be a virtual knot diagram. The virtual knot diagram obtained from D by switching the over-under information of all classical crossing (or by reflection) is denoted by D^♯ (or D^†).

Let D be a virtual knot diagram. If lk_N(D) is not zero, then D is not equivalent to D^♯ (or D^†). 


It is clear that  lk_N(D^♯)=lk_N(D^†)=-lk_N(D). This implies the result.



For example, the virtual knot presented by the diagram D in Figure <ref>, is not normal by Corollary <ref>, since lk_N(D)=-2. By Corollary <ref>,  D is not equivalent to D^♯ (or D^†).





 §.§ Linking invariants of even virtual links

H. Miyazawa K. Wada and A. Yasuhara defined an invariant of even virtual links by use of the virtual orientation. 
We give another definition of their invariant. 

Let D be a virtual link diagram. D is said to be even, if on each circle of a Gauss diagram of D there is an even number of endpoints of chords. A virtual knot diagram is even. If a virtual link diagram D is even, any virtual link diagram which is equivalent to D is even. 
Note that when we go around  each component of an even virtual link diagram, we meet an even number of virtual crossings. A normal virtual link diagram is even, and any a virtual link diagram which presents a normal virtual link is even.

Let D be an ordered unoriented even r-component virtual link diagram such that D=D^1∪…∪ D^r where D^i is a component of D. The virtual orientation is an orientation of D such that 
they switch an opposite direction at each virtual crossing of D as in Figure <ref>. 
(The orientation is not switched at classical crossings.)



We give D a virtual orientation O_v(D). 
The sum of signs of classical crossings between D^i and D^j whose overpass of D^i is denoted by λ_D(i,j) under the virtual orientation O_v(D). 


|λ_D(i,j)| is an invariant of ordered unoriented even virtual links.




The number of cut points of each component of an even virtual link diagram with a cut system is even.


Let (D,P_0) be even virtual link diagram D with the canonical cut system P_0. 
Each component of D has an even number of cut points, since we meet an even number virtual crossings when we go around each component of it. Any cut system of D is obtained from P_0 by 
cut point moves I, II and III. Then we have the same result in another cut system of D since by a cut point move II or  III  changes the number of cut points on a component of D by ± 2 or ± 4. 


Let D be an even  r-component virtual link diagram such that D=D^1∪…∪ D^r where D^i is a component of D. Suppose that P is a cut system of D.
A cut orientation is an orientation of (D, P) such that they switch an opposite direction at each cut point of D as in Figure <ref>. We denote it O_c(D,P). (For a cut orientation, the orientation is not switched at classical crossings and virtual crossings.)



Let (D,P) be even virtual link diagram D with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. 
We give D a cut orientation O_c(D,P). 
The sum of signs of classical crossings between D^i and D^j whose overpass of D^i is denoted by ν_(D, P)(i,j) under the cut orientation O_c(D,P). 


For two cut systems P and P' of D, let O_c(D,P) (or O_c(D,P')) be a cut orientation 
of (D,P) (or (D,P')). 
Then we have |ν_(D, P)(i,j)|=|ν_(D, P')(i,j)|.


Suppose that P' is obtained form P by one of cut point moves I, II and III to a component D^i (or two  components D^i and D^k) of D as in Figure <ref> (i), (ii) (or (iii)). 
The cases of a cut point moves I (or II), we give a cut orientation O_c(D,P) to (D, P) as in Figure <ref> (i) (or  (ii)) and O_c(D,P') to (D, P') as in Figure <ref> (i) (or  (ii))  (a) or (b).
In both cases, if a sign of each classical crossing c between D^i and D^j of D with respect to O_c(D,P) is ϵ,  the sign of c with respect to O_c(D,P') is ±ϵ.
The case of a cut point moves III, we give a cut orientation O_c(D,P) to (D, P) as in Figure <ref> (iii)  and O_c(D,P') to (D, P') as in Figure <ref> (iii) (a), (b), (c) or (d).
In each case, if a sign of each classical crossing c between D^i and D^j of D with respect to O_c(D,P) is ϵ,  the sign of c with respect to O_c(D,P') is ±ϵ.
Thus we have the result.





We denote |ν_(D, P)(i,j)| by |ν_D(i,j)|. 

|ν_D(i,j)| coincides to |λ_D(i,j)|. Thus 
|ν_D(i,j)| is an invariant of ordered unoriented virtual link.



Let D be an ordered unoriented even  r-component virtual link diagram such that D=D^1∪…∪ D^r where D^i is a component of D. 
We give a virtual orientation of D, O_v(D) to D. Let P_0 be the canonical cut system of D. There is a cut orientation of (D, P_0) 
whose orientation of each arc of D  corresponds to that of D in O_v(D) as in Figure <ref>.





We give an interpretation of the linking invariant |λ_D(i,j)| (=|ν_D(i,j)|) in terms of our covering method. 
Let (D, P) be an even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. 
We give an cut orientation O_c(D,P) to D. Let (D^*, P^*) be a copy of (D, P) and O_c(D^*,P^*) be the cut orientation of (D^*, P^*) such that an orientation of each arc of (D^*, P^*) is different from the corresponding arc of (D, P) in O_c(D,P).

We take the coherent double covering diagram of (D, P) in the same manner as in Figure <ref>. We call this the coherent double covering diagram with a cut orientation O_c(D,P) denoted by ϕ_O_c(D,P)(D,P). The virtual link diagram in Figure <ref> (ii) is the coherent double covering diagram of the virtual link diagram in Figure <ref> (i) with a cut orientation O_c(D,P).
Note that the orientation of the coherent double covering diagram derived from  cut orientations  O_c(D,P) and O_c(D^*,P^*).




Let (D, P) be an ordered unoriented even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. We denote the subset of the cut system P whose points are on arcs of D^i by P^i.
A component of the Gauss diagram of (D,P) corresponding a component D^i with P^i by G(D^i,P^i). 
For the Gauss diagram of (D⨿ D^*, P⨿ P^*), suppose that p_1^i, …, p_2n_i^i (or (p_1^i)^*, …, (p_2n_i^i)^*) are points in P^i such that the point p_j+1^i follows the point p_j^i along  D^i and the point (p_j^i)^* in (P^i)^*  coressponds to the point p_j^i. 
In what follows, we denote by the same symbol p_j^i (or (p_i^i)^*) for the point p_j^i of P^i (or (p_j^i)^* of (P^i)^*) and the corresponding point of G(P) (or G(P^*)). 


Let A_j^i (or (A_j^i)^*) be the arc of the Gauss diagram of G(D⨿  D^*, P⨿ P^*) between two points p_j^i and p_j+1 (or (p_j^i)* and (p_j+1^i)^*), and  the arc between two points p_2n_i^i and p_1^i (or (p_n_i^i)^* and (p_1^i)^*) is A_2n_i^i (or (A_2n_i^i)^*). Note that A_j^i corresponds to (A_j^i)^*. We also denote an arc of G(ϕ(D,P)) which corresponds to  A_j^i (or (A_j^i)^*)  of G(D⨿ D^*, P⨿ P^*) by A_j^i (or (A_j^i)^*), respectively. 





Let (D, P) be an even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. 
Then  ϕ(D,P) is a 2r-component virtual link diagram such that  ϕ(D,P)|_D^i is a 2-component link diagram D_1^i∪ D_2^i. Furthermore if an arc A_j^i is in  G(ϕ(D, P))|_D_1^i (or G(ϕ(D, P))|_D_2^i), then A_j+1^i and (A_j^i)^* are in G(ϕ(D, P))|_D_2^i 
(or G(ϕ(D, P))|_D_1^i).


The proof  is obtained by a similar argument as in Lemma <ref>. 


Let (D, P) be an ordered unoriented even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. We denote the subset of the cut system P whose points are on arcs of D^i by P^i.
For a classical  crossing c^i,j between D^i and D^j of (D, P), the classical crossing corresponding to c^i,j in ϕ_O_c(D,P)(D, P) between D_k^i and D_l^k is denoted by c^i,j_k,l (k,l∈{1,2}). 

Let c^i,j between D^i and D^j of (D, P) such that an arc of D^i is an overpass of c^i,j. Then the overpass of  a classical crossing c^i,j_k_1,l_1 in ϕ_O_c(D,P)(D, P) corresponding c^i,j,  is in  D_k_1^i. For the other classical crossing c^i,j_k_2,l_2 corresponding c^i,j,  its overpass is in D_k_2^i where k_2  k_1 and l_2 l_1 by Lemma <ref>. Then we have the following theorem.


Let (D, P) be an ordered unoriented even  r-component ordered unoriented virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D. For a cut orientation O_c(D,P) of (D, P), the absolute value of the sum of the classical crossings between D_k^i and D_l_1^j∪ D_l_2^j whose overpasses are in D^i_k in  the coherent double covering diagram ϕ_O_c(D,P)(D, P) with a cut orientation O_c(D,P) of (D, P), coincides to |ν_D(i,j)|.



Let c^i,j be a classical crossing of D between D^i and D^j whose  overpass is in D^i. One of two  classical crossings of ϕ_O_c(D,P)(D, P) corresponding to c^i,j  is between D_k^i and D_l_1^j∪ D_l_2^j, and its overpass  is in D_k^i. The sign of such a classical crossing is equal to that of c^i,j.


For an ordered unoriented even virtual link diagram D=D^1∪ D^2 with a virtual orientation in Figure <ref> (i), |λ_D(1,2)|=|λ_D(2,1)|=2. In Figure <ref> (ii), the diagram D with a cut system P and a cut orientation O_c(D,P) and an coherent double covering diagram of (D,P)  with a cut orientation O_c(D,P), ϕ_O_c(D,P)(D, P)=D_1^1∪D_2^1∪D_1^2∪D_2^2 are depicted, where D_1^i∪D_2^i correspond to a component D^i of D. The absolute value of the sum of the classical crossings between D_1^i (or D_2^i) and D_1^j∪ D_2^j whose overpasses belong to D_1^i (or D_2^i) coincides to |λ_D(i,j)| for i  j.



Let (D, P) be an ordered oriented even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D.
For a cut system P of D,   we denote a 2-component sub-diagram ϕ (D, P)|_D^i of  the coherent double covering diagram ϕ (D, P) which corresponds to D^i,  by D^i_1∪D^i_2. 
Let 
l̃k̃(D^i_k,D^j_l) be 
the linking number between D^i_k and D^j_l for k,l=1,2.

The set {l̃k̃(D^i_1,D^j_1), l̃k̃(D^i_1,D^j_2)}  is denoted by Q_ij(D)(i j). For a virtual knot diagram D, lk_N(D) is l̃k̃(D_1,D_2), where ϕ(D,P)=D_1∪ D_2.


The set Q_ij(D)  is an invariant of ordered oriented even virtual links. Furthermore 
Q_ij(D) is equal to the set {l̃k̃(D^i_2,D^j_1),l̃k̃(D^i_2,D^j_2)} and {l̃k̃(D^i_1,D^j_1),l̃k̃(D^i_2,D^j_1)}, {l̃k̃(D^i_1,D^j_2),l̃k̃(D^i_2,D^j_2)}.
The linking number l̃k̃(D^i_1,D^i_2) is an invariant of an ordered oriented even virtual link.



Let (D, P) be an ordered oriented even  r-component virtual link diagram with a cut system P such that D=D^1∪…∪ D^r where D^i is a component of D.
Let c^i,j be a classical crossing between D^i and D^j of (D, P). If one calssical crossing c^i,j_k_1,l_1 in ϕ(D, P) corresponding c^i,j is between  D_k_1^i and D_l_1^j and, then the other is a classical crossing c^i,j_k_2,l_2 between D_k_2^i and D_l_2^j for k_1 k_2 and l_1 l_2. The sings of two classical crossings c^i,j_k_1,l_1 and c^i,j_k_2,l_2 are the same. 
Then we have l̃k̃(D^i_1,D^j_1)=l̃k̃(D^i_2,D^j_2) and l̃k̃(D^i_1,D^j_2)=l̃k̃(D^i_2,D^j_1). 


Let  an ordered oriented even virtual link diagrams D=D^1∪ D^2 with a cut system (P=∅) and  D'=D^1'∪D^2' with a cut system P' be as in Figure <ref> (i). In Figure <ref> (ii) we show a coherent double covering diagram 
ϕ(D', P')=D_1^1'∪D_2^1'∪D_1^2'∪D_2^2'.
We have l̃k̃(D^1_1,D^1_2)=0 and l̃k̃(D^1_1',D^1_2')=2. So D is not equivalent to D'.







Acknowledgement

The author  would like to thank Seiichi Kamada for his useful suggestion. 

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