83

Divisibility of Lee’s class and its relation with Rasmussen’s
invariantTaketo Sano

Graduate School of Mathematical Sciences, The University of Tokyo

1

Introduction

In this article we introduce a knot invariant \overline{s}_{c} derived from the divisibility of Lee’s class,and summarize its relation with Rasmussen’s s ‐invariant. A more detailed discussion canbe found in the author’s preprint [13].
Rasmussen [12] introduced the s ‐invariant and he proved that: (i) s defines a homo‐morphism from the knot concordance group in S^{3} to 2\mathbb{Z} , (ii) it provides a lower boundfor the slice genus g_{*} of knots, and (iii) s and g_{*} are equal for positive knots. Then the
Milnor conjecture [10] follows as a corollary. The conjecture was originally proved by
Kronheimer and Mrowka in [4] using gauge theory, but Rasmussen’s result was notablesince it provides a purely combinatorial proof of an important fact of four‐dimensional
topology.

The definition of s is based on Lee homology [6] (a variant of Khovanov homology [2]).
The above stated results are obtained from the invariance of the “canonical generators” of
Lee homology over \mathb {Q} . For a knot diagram D , there are two distinct cycles \alpha, \beta constructedcombinatorially fromD,and those homology classes form a basis of H_{Lee}(D;\mathbb{Q}) . Lee

originally introduced these classes, and Rasmussen proved that they are knot invariants(up to multiplication by units), hence called them the canonical generators.

Figure 1: Construction of the cycle \alpha.

The construction of Lee’s classes can be done over \mathb {Z} , so one may expect that theyalso generate the homology group over \mathb {Z} . However this is not the case. In fact by directcomputation, we observe that many of these classes are divisible by 2‐powers. The aim ofour research is to explore what information we can obtain from the 2‐divisibility of theseclasses.

We consider this situation in a more generalized setting. By following the arguments

given by Khovanov in [3] and by Mackaay, Turner, Vaz in [9], we define a family of

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Khovanov‐type link homology theories H_{c}(-;R)_{c\in R} over an arbitrary commutative ring
R parameterized by c\in R . Here Khovanov’s original theory corresponds to c=0 , andLee’s theory corresponds to c=2 . For each c\in R , Lee’s classes [\alpha], [\beta] of a knot diagram
D can be defined in

H_{c}(D;R) .

Proposition 1. With the above setting:
1. If c is invertible in R , then [\alpha], [\beta] form a basis of H_{c}(D;R) .
2. If D, D' are two diagrams related by a single Reidemeister move, then thethe corresponding two classes is given by\pm c^{j} for some j\in\{0, \pm 1\}.

/ratio ’ ofThus the situation is completely analogous to \mathb {Q} ‐Lee theory when c is invertible. As forIn thefollowing we assume R is an integral domain and c is non‐zero, non‐invertible. We define\mathb {Z} ‐Lee theory c=2 is not invertible, so our concern is when c is not invertible in R .

the c ‐divisibility of [\alpha] (or [\beta] ) by the exponent of its c‐power factor (modulo torsions),
and denote it by k_{c}(D) . From following proposition we may regard k_{c} as measuring the“non‐positivity” of the diagram.
Proposition 2. If D is positive, then k_{c}(D)=0.

From Proposition 1 (2.) we see that k_{c} varies by the exponent j of the ratio under the
Reidemeister moves, hence k_{c} is not a knot invariant. Here j can be calculated without
computing the homology group or the homology class:
Proposition 3. The exponentj is given by the difference of \frac{r-w}{2} between the two diagrams,
where w is the writhe and r is number of Seifert circles.
If we subtract \frac{r-w}{2} from k_{c} , then the value becomes constant under the Reidemeistermoves. Hence we obtain a knot invariant:

Theorem 1. Let K be a knot. For any diagram D of K , the value\overline{s}_{c}(K) :=2k_{c}(D)+w(D)-r(D)+1
is an invariant of K.

Again by computational experiments done for (R, c)=(\mathbb{Z}, 2) , we observed that \overline{s}_{c}coincides with s for all prime knots of crossing number up to 11. Here we pose the mainquestion:

Question 1. Does \overline{s}_{c} coincide with s for any (R, c)^{\beta)}
If this is true, then s can be given a description in terms of the divisibility of Lee’sclass. If not, then we obtain some potentially new invariants. We give several facts that
support the affirmative answer.
Theorem 2.
1. \overline{s}_{c} is a 2\mathbb{Z} ‐valued knot concordance invariant in

2. |\overline{s}_{c}(K)|\leq 2g_{*}(K) for any knot K , andS^{3},

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3. \overline{s}_{c}(K)=2g_{*}(K)=2g(K) if K is positive.

These properties are common to s (except that s also possesses the homomorphism
property), and as a corollary, each \overline{s}_{c} gives an alternative proof for the Milnor conjecture.
Next, we show that there exists a pair (R, c) such that \overline{s}_{c} coincides with s.
Theorem 3. If (R, c)=(\mathbb{Q}[h], h) ,s(K)=\overline{s}_{h}(K)for any knot K.
F

More generally, we can prove that for any field F of char F\neq 2 , the s ‐invariant overand \overline{s}_{c} for (R, c)=(F[h], h) coincides. There is a famous open question whether

there exists any F such that s(-;F) is distinct from s=s(-;\mathbb{Q}) ( [8 , Question 6.1]). If
Question 1 is solved affirmatively, then it follows that s(-;F) are equal among all fields
F of char F\neq 2.

Viewing thes‐invariant from the perspective of divisibility has been suggested by

Kronheimer and Mrowka in [5], and by Collari in [1], both based on the alternative
definition of s given by Khovanov in [3]. We expect that our approach would also lead toa better understanding of s.
2

Preliminaries

In this section we briefly review the construction of Khovanov’s chain complex in thegeneralized form, as given in [3]. Let

R bea commutative ring with unity. Given a

Frobenius algebra over
C_{A}(D;R) and
its homology H_{A}(D;R) by following the construction of the original version of Khovanov
homology, except that the algebra R[X]/(X^{2}) is replaced with A . The construction is
R and a link diagram D , we obtain a chain complex

given as follows: a choice of a simultaneous resolutions for all crossings of D (which iscalled a state) yields a diagram consisting of disjoint circles. Consider all possible states of
D and place them on the vertices of an n ‐dimensional cube, so that each edge corresponds

to a cobordism between two resolved diagrams. The Frobenius algebra A determines a

TQFT (i.e. a functor from the category of 2‐dimensional cobordisms to the category of
R‐modules), and by applying it to the cube we obtain a commutative cubic diagram of
R‐modules and R ‐homomorphisms.

After some adjustment of signs of the maps, the cube

is folded to form the (unnormalized) chain complex (see Figure 2). Finally the normalizedversion is obtained after some degree shift.
Here we consider the algebra of the following special form A_{h,t}=R[X]/(X^{2}-hX-t)
with h, t\in R . We denote the corresponding chain complex and its homology by C_{h,t} and

H_{h,t} respectively. Khovanov’s original theory [2] and Lee’s theory [6] are given by
H_{Kh}=H_{0,0}, H_{Lee}=H_{0,1}.

The chain complex is also given a secondary grading called the q‐degree, and undersome condition H_{h,t}(D;R) becomes bigraded or filtered. The following theorem assuresthat any H_{h,t} gives a link invariant:

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D=

\{ 0arrow\sigma^{-}(D)arrow\overline{C}^{1}(D)arrow\overline{C}^{2}(D) arrow\overline{C}^{3}(D)arrow 0\}
Figure 2: The (unnormalized) chain complex for the left hand trefoil.

Theorem 2.1 ([3, Proposition 6]). Let L be a link. For any diagram D of L , the isomor‐
phism class of H_{h,t}(D;R) as a (graded / bigraded/filtered) R ‐module is an invariant ofL.

3

Generalizing Lee’s class

For simplicity we focus on knots, but the following discussion can be extended to links
otherwise stated. We assume that h, t\in R satisfies the following:
Condition 3.1. X^{2}-hX-t factors into linear polynomials.
Fix one c=\sqrt{h^{2}+4t} , and take the two roots:u=(h-c)/2, v=(h+c)/2\in R.
ThenX^{2}-hX-t=(X-u)(X-v) in R[X] . Define
a=X-u, b=X-v\in A.

Then obviously ab=ba=0 .

Also with a—b=v-u=c, we have:m(a\otimes a)=ca, \triangle(a)=a\otimes a,m(a\otimes b)=0, \triangle(b)=b\otimes bm(b\otimes a)=0m(b\otimes b)= −cb
where m, \triangle is the multiplication and the comultiplication of A respectively. In the case of

Lee’s theory (where (R, h, t)=(\mathbb{Q}, 0,1) ), the corresponding values are c=2,

a=X+1

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and b=X-1 . We call a and b colors. For any state s , a coloring on the resolved circles
defines an element in C_{h,t}(D;R) by the tensor product of factors in a and b . Recall thatany oriented knot diagram possesses a unique orientation preserving state s , where every
state circle admits an orientation coherent with the given orientation of D . From thestandard procedure of the bicoloring of the Seifert circles (see Figure 1), we obtain anelement \alpha\in C_{h,t}(D;R) , which is in fact a cycle. By reversing the orientation of D , weobtain another cycle \beta . (In the case of an \ell‐component link diagram D , there are 2^{\ell}
distinct cycles, one for each alternative orientation on D. )Lee proved in [6] that H_{Lee}(D;\mathbb{Q}) is freely generated by these classes, so in particular

H_{Lee}(D;\mathbb{Q})\cong \mathbb{Q}^{2} . This generalizes as follows:

Proposition 3.2. If c=\sqrt{h^{2}+4t} is invertible in R , then H_{h,t}(D;R) is freely generatedby [\alpha], [\beta] over R. In particular H_{h,t}(D;R)\cong R^{2}.
Lee’s proof cannot be applied directly, since it uses Hodge theory and requires that R
is a field. However there is an alternative proof that can be applied to our case, that is

the admissible coloring decomposition of C_{h,t}(D;R) proposed by Wehrli in [15, Remark
5.4]. Briefly, if c is invertible, then the colored states form a basis of C_{h,t}(D;R) , and there
is a decomposition of C_{h,t}(D;R) by the set of admissible colorings of D . The subcomplex

spanned by \alpha, \beta gives the homology, while the remaining part is acyclic. A detailed proof

can be found in Lewark’s paper [7, Lemma I.14].
The following proposition is a generalization of [12, Proposition 2.3], and is essential
for the definition of the invariant \overline{s}_{c} in the next section.

Proposition 3.3. Suppose D, D' are two diagrams related by a Reidemeister move. Thereis an isomorphism \rho : H_{h,t}(D;R)arrow H_{h,t}(D';R) such that the \alpha, \beta ‐classes correspond as:[\alpha']=\varepsilon c^{j}\rho[\alpha],[\beta']=\varepsilon'd\rho[\beta]
with some j\in\{0, \pm 1\} and \varepsilon, \varepsilon'\in\{\pm 1\} satisfying \varepsilon\varepsilon'=(-1)^{j} . (Here c is not necessarily
invertible, so the equation z=c^{j}w is to be understood as c^{-j}z=w when j<0. ) Moreover
j is determined as in Table 1 by the type of the move and the difference of the numbersof Seifert circles.
Type\triangle r|j\frac{RM1_{R}1RM1_{L}1 _{1}^{0}{RM20, 1_{ }^{0}
RM3

0

0

2

1

-2

-1

Table 1: The exponent j corresponding to the Reidemeister moves

88This is proved by considering all possible patterns that occur under each move. In
particular if c is not invertible, then [\alpha], [\beta] are not invariant under the Reidemeistermoves. From Table 1 we see that j is given by a single equation:
Corollary 3.4. The exponent j is given by

j= \frac{\triangle r-\triangle w}{2}wheredenotes the writhe,denotes the number of Seifert circles, and the prefixed \triangle
denotes the difference of the corresponding values of D and D'.
wr

Finally we state that under Condition 3.1 the two parameters h, t\in R can be reduced
to a single parameter c\in R.
Proposition 3.5. Let (h, t), (h', t') be two pairs satisfying \sqrt{h^{2}+4t}=\sqrt{h^{\prime 2}+4t'} . For
any link diagram D , there is an isomorphism \sigma : H_{h,t}(D;R)arrow H_{h',t'}(D;R) that commuteswith the isomorphism \rho of Proposition 3.3, and that the \alpha, \beta ‐classes correspond one‐to‐one.

We denote the isomorphism class of H_{h,t}(D;R) by H_{c}(D;R) . Lee’s classes [\alpha], [\beta] arewell‐defined under the identification. Figure 3 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 tothe isomorphism class H_{c}(D;R) .

Figure 3: The (h, t) ‐parameter space.

From the above propositions, we conclude that the situation is completely analogous to\mathb {Q} ‐Lee theory when c is invertible: [a] and [\beta] form a basis of H_{c}(D;R) and are invariant(up to unit) under the Reidemeister moves. Now our main concern is when c is notinvertible in R.

4

Divisibility of Lee’s class

In the remaining we assume that R is an integral domain and c is non‐zero, non‐invertible.
Denote by H_{c}(D;R)_{f} the free part of H_{c}(D;R) , i.e. the quotient of H_{c}(D;R) by its torsion
submodule. By abuse of notation, we denote the image of an element [z]\in H_{c}(D;R) by
the same symbol [z]\in H_{c}(D;R)_{f} . We define the c ‐divisibility of [\alpha] by the exponent of

its c‐power factor (modulo torsions). More formally,

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Definition 4.1. For any knot diagram D , definek_{c}(D)= \max\{k\geq 0|[\alpha]\in c^{k}\cdot H_{c}(D;R)_{f} \} \in[0, \infty] .
Example 4.2. k_{c}(O)=0 since H(O)=c(O)=R\langle 1, X }.
Example 4.3.

D=

(unknot with one negative crossing). Let (R, h, t)=(\mathbb{Z}, 0,1) and

c=2 . We have

C(D)= \{ Aarrow^{\triangle}A^{\otimes 2} \}, \alpha(D)=(X-1)\otimes(X+1).

From \triangle(X)=X\otimes X+1\otimes 1 and \triangle(1)=1\otimes X+X\otimes 1 , we see that \alpha(D) is homologous
to 2(1\otimes X-1\otimes 1) . Since \{[1\otimes X], [1\otimes 1]\} form a basis of H(D)_{f} , we have k_{2}(D)=1.
These examples show that k_{c}(D) is not a link invariant. In fact from Proposition 3.3,
the difference of k_{c} between the two diagrams can be given without even computing thehomology groups.

Proposition 4.4. Let D, D' be two diagrams of the same knot. Then

\triangle k_{c}=\frac{\triangle r-\triangle w}{2},where w denotes the writhe, r denotes the number of Seifert circles, and the prefixed \triangle
denotes the difference of the corresponding values of D, D'.
Thus we obtain a knot invariant:

Theorem 1. For any knot K,\overline{s}_{c}(K)=2k_{c}(D)+w(D)-r(D)+1,
gives an invariant of K , where D is any diagram of K.
First we state some basic properties of k_{c} . The following ones can be proved byobservations of the diagram with some elementary algebraic arguments.
Proposition 4.5. For any knot diagram D

0\leq k_{c}(D)\leq n^{-}(D),where n^{-}(D) denotes the number of negative crossings. In particular if D is positive, thenk_{c}(D)=0.

Thus we may regard k_{c} as measuring the “non‐positivity” of the diagram.
Proposition 4.6.

k_{c}(D)=k_{c}(-D).

The above proposition tells us that we may either use [\alpha] or [\beta] for the definition ofk_{c}(D).

Proposition 4.7. Let D, D' be knot diagrams.

901.

k_{c}(D\sqcup D')\geq k_{c}(D)+k_{c}(D') .

2.

k_{c}(D\# D')\leq k_{c}(D\sqcup D')\leq k_{c}(D\# D')+1.

Moreover, if R is a PID and c is prime in R , then instead we have:1’k_{c}(DuD')=k_{c}(D)+k_{c}(D') .

2’k_{c}(D\# D')=k_{c}(D)+k_{c}(D')

k_{c}(D)+k_{c}(D')-1.

or

Next we state some basic properties of \overline{s}_{c} . The following can be obtained immediatelyfrom the definition and the corresponding results for k_{c}.
Proposition 4.8. Let K, K' be any two knot diagrams.
1.

\overline{s}_{c}(O)=0.

2.

\overline{s}_{c}(K)=\overline{s}_{c}(-K) .

3.

\overline{s}_{c}(KuK')\geq\overline{s}_{c}(K)+\overline{s}_{c}(K')-1.

4. \overline{s}_{c}(K\# K')=\overline{s}_{c}(K\sqcup K')\pm 1.

Moreover if R is a PID and c is prime in R , then we have3’.

\overline{s}_{c}(K\sqcup K')=\overline{s}_{c}(K)+\overline{s}_{c}(K')-1.

4’. \overline{s}_{c}(K\# K')=\overline{s}_{c}(K)+\overline{s}_{c}(K')

or \overline{s}_{c}(K)+\overline{s}_{c}(K')-2.

Proposition 4.9.

\overline{s}_{c}(K)\equiv 0mod 2.
Proposition 4.10. Let K be a positive knot, and D be a positive diagram of L. Let S be
the Seifert surface of K obtained by applying the Seifert’s algorithm to D. Then\overline{s}_{c}(K)=2g(S).

The above two propositions easily follows from\chi(S)=1-2g(S)=r(D)-n(D).

Remark 4.11. We have modded out the torsions so that Proposition 4.4 holds. In general,
if there is a torsion, then c‐divisibility is not additive under multiplying a power of c.
Consider the case (R, c)=(\mathbb{Z}, 2), H=\mathbb{Z}\oplus \mathbb{Z}_{2} , and z=(2,1) . In this case k_{2}(z)=0,
but 2z=(4,0) so k_{2}(2^{1}z)=2 . The c ‐invariant of a transverse link is introduced by

Collari in [1], whose definition is similar to k_{c}(D) but it is defined without modding out

the torsions.

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91
5

Behavior of \overline{s}_{c} under cobordisms

Let K, K' be two knots in \mathb {R}^{3} and D, D' be diagrams of K, K' respectively. Let S\subset\mathbb{R}^{3}\cross[0,1] be an (oriented smooth) cobordism between K and K' with \partial S=(-K)\cross\{0\}\cup K'\cross\{1\} . By following the arguments of [2, Section 6.3] and [12, Section 4], we

define a homomorphism\phi : H_{c}(D;R)arrow H_{c}(D';R)

as follows. By modifying S by a small isotopy, we may assume that S decomposes as aunion of elementary cobordisms, such that for each t\in[0,1] the section S\cap(\mathbb{R}^{3}\cross\{t\}) is alink and its projection onto the plane is regular, except for finitely many t' s . Decompose
S= \bigcup_{i=0}^{N-1}T_{i} so that each T_{i}=S\cap(\mathbb{R}^{3}\cross[t_{i}, t_{i+1}]) corresponds to a Reidemeister moveor a Morse move. Denote by D_{i} the projection of the link S\cap(\mathbb{R}^{3}\cross\{t_{i}\}) . Each T_{i} gives
a homomorphism \phi_{i} : H_{C}(D_{i})arrow H_{C}(D_{i+1}) , namely if T_{i} corresponds to a Reidemeistermove then \phi_{i} is the isomorphism \rho given in Proposition 3.3, and if T_{i} corresponds to a
Morse move, then \phi_{i} is induced from the chain map given by the corresponding operationof the Frobenius algebra A . Define \phi by the composition of all \phi_{i}' s . The following is a

generalization of [12, Proposition 4.1] and [11, Proposition 3.2].
Proposition 5.1. If S is a connected cobordism between K and K' , then \phi maps

\phi[\alpha(D)]=\pm c^{l}[\alpha(D')] , l=\frac{1}{2}(-\triangle r+\triangle w -\chi(S),where the prefixed \triangle denotes the difference of the corresponding values of D, D'.
Proof. This is proved by carefully inspecting the successive images of [\alpha(D)] under themaps \phi_{i} . Figure 4 depicts the schematic picture. Here we may assume that c is invertible,
since the canonical map H_{c}(D;R)_{f}arrow H_{c}(D;R)_{f}\otimes R[c^{-1}]=H_{c}(D;R[c^{-1}]) is injective.
L'[a_{4}]\ovalbx{tsmREJCT}

c^{-1}m\Deltaio[_{3}]1

0\Delta]\rho 1

L[a_{0}]

Figure 4: The cobordism map\squareWe immediately obtain:

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Proposition 5.2. Proposition 5.1,|\overline{s}_{c}(K')-\overline{s}_{c}(K)|\leq-\chi(S).

From Proposition 5.2 we obtain properties of \overline{s}_{c} that are common to s . From theseproperties, the Milnor conjecture can be reproved using each of \overline{s}_{c}.
Theorem 2.

1. \overline{s}_{c} is a knot concordance invariant,

2. |\overline{s}_{c}(K)|\leq 2g_{*}(K) for any knot K , and3. \overline{s}_{c}(K)=2g_{*}(K)=2g(K) if K is positive,
where g_{*} is the slice genus, and g is the ordinary genus of a knot.

Corollary 5.3 (The Milnor Conjecture [10]). The slice genus and the unknotting numberof the (p, q) torus knot are both equal to (p-1)(q-1)/2.
6

Coincidence withs

Finally we prove that \overline{s}_{c} coincides with s for the case (R, c)=(\mathbb{Q}[h], h) . We declare thath is an indeterminate of degree -2 , so that the corresponding homology group H_{h}(D;R)
becomes bigraded. Recall that in the case of (R, c)=(\mathbb{Q}, 2) , Rasmussen called thecorresponding classes [\alpha], [\beta] the “canonical generators from the fact that they form abasis and that they are invariant (up to unit) under the Reidemeister moves. We haveseen that this does not hold for a general (R, c) . However for (R, c)=(\mathbb{Q}[h], h) , by“normalizing” the two classes we obtain a pair of classes [\zeta], [\zeta'] that are reasonable to becalled the “canonical generators” of H_{h}(D;\mathbb{Q}[h])_{f}.
Proposition 6.1. There is a unique pair of classes [\zeta], [\zeta']\in H_{h}(D;\mathbb{Q}[h])_{f} such that:\bulletthey form a basis of H_{h}(D;\mathbb{Q}[h])_{f},\bulletthey are (strictly) invariant under the Reidemeister moves, and,\bullet[\alpha], [\beta] can be written as[\alpha]= h^{k}( (h/2)[\zeta]+[\zeta'])[\beta]=(-h)^{k}(-(h/2)[\zeta]+[\zeta'])wherek=k_{h}(D) .

Note that from the descriptions of [\alpha], [\beta] , it is obvious that those h ‐divisibility isgiven by k . Here we only state that the proof is non‐constructive, and it depends on thealgebraic property of R=\mathbb{Q}[h] . Also this result is only obtained for knots at the time ofwriting.
Denote by D^{*} the mirror image of D . There is a unimodular pairing\langle-, -\rangle:C_{h}(D)\otimes C_{h}(D^{*})arrow R

93
defined by the composition of the natural identification T : C(D^{*})arrow C(D)^{*} (the dual
complex of C(D) ) and the standard pairing between C(D) and C(D)^{*} . From a generalargument of homological algebra, this descends to\langle-, -\}:H_{h}(D)_{f}\otimes H_{h}(D^{*})_{f}arrow R,
and is unimodular sinceLemma 6.2. Let

R=\mathbb{Q}[h] is a PID.

(\alpha, \beta)=(\alpha(D), \beta(D)) and (\alpha^{*}, \beta^{*})=(\alpha(D^{*}), \beta(D^{*})) .

\langle (\begin{ar y}{l \alph \beta \end{ar y}), (\alpha^{*} \beta^{*})\rangle=h^{r(D)} (\begin{ar y}{l \pm1 0 0 \pm1 \end{ar y})

Then

.

Proof. Froma=X+(h/2)1 , b=X-(h/2)1
T(a)=1^{*}+(h/2)X^{*}, T(b)=1^{*}-(h/2)X^{*}
we have:\langle (\begin{ar y}{l a b \end{ar y}), (T(a) T(b) \rangle=(\begin{ar ay}{l } c 0 0 -c \end{ar ay})
The result follows since the Seifert circles of D and D^{*} are identical, and from the con‐\squarestruction of the cycles \alpha, \beta.

Proposition 6.3 (Mirror formula).
k_{h}(D)+k_{h}(D^{*})=r(D)-1.
Proof. With the description of Proposition 6.1,

=h^{k+k^{l}}\langle (\begin{ar y}{l \alph \beta \end{ar y}), (\alpha^{*} \beta^{*})\rangle(\begin{ar ay}{l} h/2 1 \mph/2\pm1 \end{ar ay}) \langle (\begin{ar y}{l \zeta \zeta \end{ar y}), (\zeta^{*},\zeta^{\prime*})\rangle(\begin{ar ay}{l} h/2 \mph/2 1 \pm1 \end{ar ay})

Since the pairing is unimodular, the middle matrix on the right hand side must haveunital determinant. By comparing the determinants on both sides we have2r(D)=2(k+k')+2.
\squareProposition 6.4. Let D, D' be knot diagrams.
k_{h} ( D # D ’)

=k_{h}(D)+k_{h}(D') .

Proof. From Proposition 4.7 we havek_{h} ( D # D ’)\leq k_{h}(D)+k_{h}(D') .

94
Together with Proposition 6.3 we obtain the reverse inequality:k_{h} ( D # D ’)=-k_{h}((D\# D')^{*})+r((D\# D')^{*})-1\geq-(k_{h}(D^{*})+k_{h}(D^{\prime*}))+(r(D)+r(D')-1)-1=k_{h}(D)+k_{h}(D').

\squareProposition 6.5. \overline{s}_{h} defines a homomorphism from the concordance group of knots inS^{3} to 2\mathbb{Z}.

Proof. Immediate from Proposition 6.3, 6.4.

\squareNow we are ready to prove the main statement:
Theorem 3. For any knot K,

s(K)=\overline{s}_{h}(K;\mathbb{Q}[h]).

Proof. Since both s and \overline{s}_{h} changes sign by mirroring the knot, it suffices to prove theinequalitys(K)\geq\overline{s}_{h}(K).

Let \alpha, \alpha_{h} be the \alpha ‐cycles of D in C_{Lee}(D;\mathbb{Q})=C_{2}(D;\mathbb{Q}) and in C_{h}(D;\mathbb{Q}[h]) respec‐tively. Then\pi:h\mapsto 2

maps \pi(\alpha_{h})=\alpha . Let [\alpha_{h}]=h^{k}[\alpha_{h}'] with maximal k . Obviously

q\deg_{h}[\alpha_{h}']=2k+q\deg_{h}[\alpha_{h}]=2k+w(D)-r(D).

Recall that H_{h}(D;\mathbb{Q}[h]) is bigraded, whereas H_{Lee}(D;\mathbb{Q}) is only filtered. Since \pi isq‐degree non decreasing, we haves(K)= qdeg [\alpha]+1=q\deg(\pi_{*}[\alpha_{h}])+1=q\deg(\pi_{*}[\alpha_{h}'])+1\geq q\deg_{h}[\alpha_{h}']+1=\overline{s}_{h}(K;\mathbb{Q}[h]).

\squareRemark 6.6. There is a well known lower bound for s ([14, Lemma 1.3])
s(K)\geq w(D)-r(D)+1,
so we see that 2k_{h}(D) gives the correction term of the inequality.
Corollary 6.7.

s(K)=q\deg_{h}[\zeta]-1=q\deg_{h}[\zeta']+1.

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Proof. Obvious from Theorem 3 and the description of [\alpha] given in Proposition 6.1.

\squareCompared to the definition of s given by the filtration of \mathb {Q} ‐Lee homology, this gives a
direct characterization of s by the canonical generators [\zeta], [\zeta'].
Remark 6.8. The above discussion works verbatim for any (R, c)=(F[h], h) when F isa field of char F\neq 2 . On the other hand there is a generalization of s over an arbitrary
field F[8] . Theorem 3 generalizes ass(K;F)=\overline{s}_{h}(K;F[h]) ,char F\neq 2.

We end this article with some open questions.

Question 6.9. Is there any (R, c) such that \overline{s}_{c}(-;R) is distinct from s ? Or are they all
equal?
It is a famous open question whether there exists any F such that s(-;F) is distinct
from s=s(-;\mathbb{Q}) ( [8 , Question 6.1]). Theorem 3 implies that if Question 6.9 is solved
affirmatively, then all s(-;F) are equal among fields F of char F\neq 2.

Remark 6.10. As for F=\mathbb{F}_{2} , Seed showed by direct computation that

K=K14n19265

has s(K;\mathbb{Q})=0 but s(K;\mathbb{F}_{2})=-2 (see [8, Remark 6.1]).
Question 6.11. Does the (refined) canonical generators [\zeta], [\zeta'] exist for a general (R, c) ?
Is there any geometric explanation for these classes?
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Graduate School of Mathematical Sciences

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E‐mail address: tsano@ms.u‐tokyo.ac.jp