Divisibility of Lee's class and its relation with Rasmussen's invariant
Taketo Sano


    Fast fabrication of optical vortex generators by femtosecond laser ablation
    Jian-Guan Hua^1, Zhen-Nan Tian^1^*, Si-Jia Xu^1, Stefan Lundgaard^2,3,
Saulius Juodkazis^2,3^*
    December 30, 2023
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    Lee homology (a variant of Khovanov homology) over  possesses the “canonical generators" as its basis. The generators (Lee's classes) [(D, o)] are constructed combinatorially from an oriented link diagram D, one for each alternative orientation o on D. Let R be an integral domain. There exists a family of link homology theory { H_c(-; R) }_c ∈ R, where Khovanov's theory corresponds to c = 0 and Lee's theory corresponds to c = 2. For each c ∈ R ∖ 0, Lee's classes [(D, o)] can be defined as elements in H_c(D; R), but when c is not invertible then they do not form a basis; in fact they are divisible by c-powers. We define the c-divisibility k_c(D) of [(D, o)] with o the given orientation of D. For any link L and its diagram D, we prove that s̅_c(L) := 2k_c(D) + w(D) - r(D) + 1 is a link invariant, where w is the writhe, and r is the number of Seifert circles. We pose the question whether s̅_c coincides with Rasmussen's s-invariant. There are several evidences that support the affirmative answer. For instance, _c is a link concordance invariant, and the Milnor conjecture can be reproved using _c. Also for the special case (R, c) = ([h], h), our s̅_c actually coincides with s as knot invariants.


    

§ INTRODUCTION


Almost two decades have passed since Khovanov introduced in <cit.> a link homology theory, now known as Khovanov homology, that categorifies the Jones polynomial. In <cit.> Rasmussen introduced a knot invariant s based on Lee homology (a variant of Khovanov homology introduced by Lee in <cit.>). He proved: (1) the invariant s defines a homomorphism from the knot concordance group in S^3 to 2, (2) it provides a lower bound for the slice genus g_* of knots, and (3) s and g_* are equal for positive knots. Then the Milnor conjecture <cit.> follows as a corollary. The conjecture was originally proved by Kronheimer and Mrowka in <cit.> using gauge theory, but Rasmussen's result was notable since it provided for the first time a purely combinatorial proof.



The well-definedness of s is based on the invariance of the “canonical generators" of Lee homology over . Let L be an oriented link and D be its diagram. The generators [(D, o)] ∈ H_(D; ) are constructed combinatorially from D, one for each alternative orientation o on D. Lee introduced these classes in <cit.> and proved that they form a basis of H_(D; ). Rasmussen then proved that they are invariant (up to unit) under the Reidemeister moves, hence called them the canonical generators of H_(L; ). The construction can be done over , so one may expect that they form a basis of H_(D; ). However, by direct computation, we see that there are many diagrams such that the classes are divisible by 2-powers (hence do not form a generating set), and the 2-divisibility is not constant among diagrams of the same link. Where does `2' come from? How does 2-divisibility of the classes vary under the Reidemeister moves?



In this paper, we show that `2' is the difference of the two roots of the polynomial X^2 - 1 which is used in the construction of Lee homology. This phenomenon can be seen in a more generalized setting. Khovanov introduced in <cit.> a family of link homology theories { H_h, t(-; R) }_h, t ∈ R over an arbitrary commutative ring R. Khovanov's original theory is given by (h, t) = (0, 0) and Lee's theory is given by (h, t) = (0, 1). From the arguments given by Mackaay, Turner, Vaz in <cit.>, we see that if the polynomial X^2 - hX - t factors into linear polynomials, then Lee's classes [(D, o)] can be defined in H_h, t(D; R).


    With the above setting, let c = √(h^2 + 4t).
    
        
  * If c is invertible in R, then Lee's classes form a basis of H_h, t(D; R). (<Ref>)
        
        
  * If D, D' are two diagrams related by a single Reidemeister move, then the `ratio' of the corresponding pair of classes [(D, o)] and [(D', o')] is given by ± c^j for some j ∈{0, ± 1}. (<Ref>)
        
        
  * For another (h', t') such that c = √(h'^2 + 4t'), the corresponding groups H_h, t(D; R) and H_h', t'(D; R) are naturally isomorphic, and under the natural isomorphism Lee's classes correspond one-to-one. (<Ref>)
    


The first two statements imply that the situation is completely analogous to -Lee theory when c is invertible. The third says that the isomorphism class of H_h, t(D; R) is determined by c, and Lee's classes are well-defined under the identification. Thus we obtain a family of link homology groups {H_c(D; R)}_c ∈ R. <Ref> depicts the (h, t)-parameter space, where each point (h, t) corresponds to H_h, t(D; R) and the parabola h^2 + 4t = c^2 corresponds to the isomorphism class H_c(D; R). 



Next we assume R is an integral domain, and consider when c is non-zero, non-invertible in R. In this case, Lee's classes do not form a basis of H_c(D; R); in fact they are divisible by c-powers. Let o be the given orientation of D. We define the c-divisibility of [(D, o)] (modulo torsions) by the exponent of its c-power factor, and denote it by k_c(D). From the following proposition, we may regard k_c as measuring the “non-positivity" of the diagram. 


    If D is positive, then k_c(D) = 0. (<Ref>)


In another paper <cit.> of Rasmussen's, the variance of the canonical generators under the Reidemeister moves is stated in more detail. The statement is generalized to our case as follows:


    The variance of k_c under each Reidemeister move is equal to the variance of r - w/2, where w is the writhe, and r is number of Seifert circles. (<Ref>)


Thus we obtain:


    Let L be a link. For any diagram D of L, the value
    
    _c(L) := 2k_c(D) + w(D) - r(D) + 1

    is an invariant of L.


From the computational experiments[
    <https://git.io/fphro>
] 
done for (R, c) = (, 2) (i.e. -Lee theory), we see that the values of _2 coincide with those of s for prime knots of crossing number up to 11. Here we pose the main question:

 
    Does _c coincide with s for any (R, c)?


We give several theoritical evidences that support the affirmative answer. First, by inspecting the behavior of _c under cobordisms, we obtain properties of _c that are common to s:

 
    
        
  * _c is a link concordance invariant,
        
  * |_c(K)| ≤ 2g_*(K) for any knot K, and
        
  * _c(K) = 2g_*(K) = 2g(K) if K is positive.
    


As a corollary, the Milnor conjecture can be reproved using _c. Next, we show that there exists (R, c) such that _c coincides with s as knot invariants. The original definition of s uses -Lee homology, but it can be generalized over an arbitrary field F of F≠ 2. For the special case (R, c) = (F[h], h) where h is an indeterminate of degree -2 (this corresponds to the Bar-Natan theory over F), we prove:

 
    For any knot K, 
    
    s(K; F) = _h(K; F[h])

    where s(K; F) is the s-invariant of K over F.


It is a famous open question whether there exists any F such that s(-; F) is distinct from s = s(-; ) (<cit.>). <Ref> implies that if <Ref> is solved affirmatively, then all s(-; F) are equal among fields F of F≠ 2.



Viewing the s-invariant from the perspective of divisibility has been suggested by Kronheimer and Mrowka in <cit.>, and by Collari in <cit.>, both based on the alternative definition of s given by Khovanov in <cit.>. We expect that our approach would also lead to a better understanding of s.



 §.§ Outline


In Section 2, we first review the construction of Khovanov homology and Lee's classes in the generalized setting. Then we state the variance of the classes under the Reidemeister moves. In Section 3, we define k_c(D) for a link diagram D and the invariant _c(L) for a link L. We also state the behavior of _c under cobordisms, and obtain properties common to Rasmussen's s-invariant. Then we focus on knots and specialize to (R, c) = (F[h], h) where F is a field of F≠ 2, and prove that _h coincides with s over F. The final section gives further remarks and questions. 



 §.§ Conventions


In this paper, we assume knots and links are oriented. For a link L, we denote by |L| the number of components of L, by -L the link obtained from L by reversing the orientation on each of its components, and by L̅ the mirror image of L. We use the same notations for link diagrams.



 §.§ Acknowledgements


This paper is based on the master's thesis submitted to the Graduate School of Mathematical Sciences, the University of Tokyo. I am deeply grateful to my supervisor Mikio Furuta for his guidance and for everything he taught me in mathematics. I am also grateful to Kouki Sato for devoting a large amount of time for helpful discussions. I thank Yukio Kametani, Takayuki Kobayashi, Ryusuke Horiuchi for introducing me to the subject; Yoshihiro Matsumori for constructive feedbacks and correction of errors; members of swift-developers-japan for helping me improve the computer program; and the members of my academist fan club [<https://taketo1024.jp/supporters>] for the financial support. Finally, I thank my wife and my daughter for their support and encouragement throughout the years of my study and research.


    

§ KHOVANOV HOMOLOGY AND LEE'S CLASSES
 



 §.§ Khovanov homology
 

In this section, we review Khovanov homology theory in the generalized form as given in <cit.>. Let R be a commutative ring with unity. A Frobenius algebra over R is a quintuple (A, m, ι, Δ, ϵ) such that: (1) (A, m, ι) is an associative R-algebra with multiplication m and unit ι, (2) (A, Δ, ϵ) is a coassociative R-coalgebra with comultiplication Δ and counit ϵ, and (3) the Frobenius relation holds: 

    Δ∘ m = (𝕀⊗ m) ∘ (Δ⊗𝕀) = (m ⊗𝕀) ∘ (𝕀⊗Δ).


Let h, t be two elements of R. Define A_h, t = R[X]/(X^2 - hX - t) with the obvious R-algebra structure. Define the counit ϵ: A_h, t→ R by


    ϵ(1) = 0,  ϵ(X) = 1.


Then the comultiplication Δ is uniquely determined so that (A_h, t, m, ι, Δ, ϵ) becomes a Frobenius algebra. Explicitly, Δ is given by


    Δ(1) = X ⊗ 1 + 1 ⊗ X - h1 ⊗ 1,   Δ(X) = X ⊗ X + t 1.


Given a link diagram D, we obtain a chain complex C_h, t(D; R) and its homology H_h, t(D; R) by following the construction of the original Khovanov homology, except that the Frobenius algebra R[X]/(X^2) is replaced with A_h, t. The chain complex is also given a secondary grading, and under some condition H_h, t(D; R) becomes bigraded or filtered. Here we remark that Khovanov's original theory (<cit.>) and Lee's theory (<cit.>) are given by

    H_ = H_0, 0,   
        H_ = H_0, 1,

and Bar-Natan's theory (<cit.>) is given by

    H_ = H_h, 0

where h is an indeterminate of degree -2. By collapsing H_ with h = 1 we obtain the filtered version:

    H_𝑓𝐵𝑁 = H_1, 0.


The following theorem assures that any H_h, t gives a link invariant:

 
    Let L be a link. For any diagram D of L, the isomorphism class of H_h, t(D; R) as a (graded / bigraded / filtered) R-module is an invariant of L.


In order to fix the terms and notations used in the later sections, we briefly review the construction of the chain complex C_h, t(D; R). Let D be a link diagram with n crossings. Each crossing admits a 0-, 1-resolution as depicted in  <Ref>. 



A simultaneous choice of resolutions for all crossings of D is called a state. For each state s, denote by |s| the number of 1-resolutions in s. Two states are adjacent if one is obtained from the other by changing the resolution of a single crossing. We write s ≺ s' when s and s' are adjacent and |s| + 1 = |s'|. Any state s yields a diagram consisting of disjoint circles by resolving all crossings accordingly. We call such circles s-circles and denote its number by r(D, s). Each state s corresponds to an R-module V(D, s) = A^ r(D, s). An element in V(D, s) of the form:

    x = e_1 ⋯ e_r,
           e_i ∈{ 1, X }

is called an enhanced state. For each pair of adjacent states s ≺ s', there is an R-module homomorphism V(D, s) → V(D, s'), given by the multiplication m or the comultiplication Δ of A, depending on whether the s-circle(s) merge or split when the resolution of the single crossing is changed from 0 to 1. These modules and maps form a commutative n-dimensional cubic diagram. After some adjustment of signs of the maps, the cube is folded to form the unnormalized chain complex

    C̅_h, t^i(D; R) = ⊕_|s| = i V(D, s)

(see <Ref>). Then C_h, t(D) is defined by shifting the homological degree of C̅_h, t(D) by -n^-, the number of negative crossings of D. Each enhanced state x is also endowed a secondary degree, which we call the q-degree. Let (1) = 0, (X) = -2. Define (x) := ∑_i (e_i) and

    (x) := (x) + |s| + r(D, s) + n^+ - 2n^-,

where n^+, n^- are the number of positive, negative crossings of D respectively. This gives a bigrading on C_h, t(D). If R is graded with h, t having 

    (h = -4  or  h = 0)  and 
        (t = -2  or  t = 0),

then the differential d preserves the q-degree and H_h, t(D; R) inherits the bigrading. Otherwise if 

    (h≥ -4  or  h = 0)  and 
        (t≥ -2  or  t = 0),

then d is q-degree non-decreasing, so we may define a filtration on C_h, t(D; R) as

    F^j C_h, t^i(D; R) := { x ∈ C_h, t^i(D; R) |(x) ≥ j }.

H_h, t(D; R) admits the induced filtration, where the (filtered) q-degree of a homology class γ is given by the maximum q-degree among its representatives

    (γ) := max{(x) | [x] = γ}.

Note that H_ and H_ are bigraded, whereas H_ and H_f are filtered. 





Finally we state some basic properties of C_h, t that are given in <cit.>.

 
	C_h, t^i, j(D; R) is supported only where j ≡ |D| 2. 


 
     
    
        
  * C_h, t(-D) = C_h, t(D)
        
  * C_h, t(D ⊔ D')  C_h, t(D) ⊗ C_h, t(D')
        
  * C_h, t(D̅)  C_-h, t^*(D)
    
    where -D is D with its orientation reversed, and D̅ is the mirror image of D.
























    

 §.§ Lee's classes


In <cit.> Lee constructed a set of classes [(D, o)] ∈ H_(D; ), one for each alternative orientation o on D, and proved that they form a basis. By following the arguments given by Mackaay, Turner, Vaz in <cit.>, these classes can be generalized as elements in H_h, t(D; R). Throughout this section, we assume the following condition holds: 

 
	There exists c ∈ R such that h^2 + 4t = c^2 and (h ± c)/2 ∈ R.


Fix one c = √(h^2 + 4t), and let 

    u = (h - c)/2,   
        v = (h + c)/2  ∈ R.

Then X^2 - hX - t factors as (X - u)(X - v) in R[X]. Also let 

    = X - u,  =̱ X - v  ∈ A.

Then obviously =̱= 0. Also with - =̱ v - u = c, we have:


	m(⊗)    = c, 
	        
	     Δ()    = ⊗, 

	m(⊗)̱    = 0, 
	       Δ()̱    = ⊗̱
	m(⊗̱)    = 0       

	m(⊗̱)̱    = -c   


Here we call  and $̱colors. For any states, a coloring on thes-circles defines an element inV(D, s), which we call a colored state. Recall that a link diagram possesses a unique orientation preserving states, where every state circle admits an orientation coherent with the given orientation ofD. Such a state can be obtained by 0-resolving the positive crossings, and 1-resolving the negatives as in <Ref>. The corresponding state circles are the Seifert circles. We color the Seifert circles according to the following algorithm: