Khovanov homology and exotic $4$-manifolds

For a knot $K\subset S^{3}$, let $TB(K)$ denote its maximal Thurston-Bennequin number. We have the following vanishing criterion for the Khovanov skein lasagna module.

For example, since $S^{2}\times S^{2}$ contains an embedded sphere with self-intersection $2$, we see the $2$-trace of the unknot embeds into $S^{2}\times S^{2}$, implying $\mathcal{S}_{0}^{2}(S^{2}\times S^{2})=0$. This answers a question of Manolescu (see [sullivan2024kirby, Question 2]), generalizing one main Theorem of Sullivan-Zhang [sullivan2024kirby, Theorem 1.3] and answering [sullivan2024kirby, Question 3] in the affirmative.

The folklore trace embedding lemma says that $X_{n}(K)$ embeds into $X$ if and only if $-K$ is $(-n)$-slice in $X$, i.e. there is a framed slice disk in $X\backslash int(B^{4})$ of the $(-n)$-framed knot $-K\subset S^{3}=\partial B^{4}$. Therefore, the contrapositive of Theorem 1.4 can be phrased as follows.

In the construction of the Khovanov skein lasagna module, the relevant TQFT (the Khovanov-Rozansky $\mathfrak{gl}_{2}$ homology) enters only at the very last step. We could replace this TQFT by its Lee deformation [lee2005endomorphism], and define an analogous notion of a Lee skein lasagna module of a $4$-manifold $X$ with link $L\subset\partial X$, denoted $\mathcal{S}_{0}^{Lee}(X;L)$. It is a $\mathbb{Q}$-vector space equipped with a homological $\mathbb{Z}$-grading, a quantum $\mathbb{Z}/4$-grading, a homology class grading by $H_{2}(X,L)$, and a quantum $\mathbb{Z}$-filtration (where $0$ has filtration degree $-\infty$), with the caveat that a nonzero vector can have quantum filtration degree $-\infty$.

We show in Section 4 that the structure of $\mathcal{S}_{0}^{Lee}(X;L)$, except for the quantum filtration, is determined by simple algebraic data of $(X,L)$. Moreover, we extract numerical invariants from the quantum filtration structure³³3The authors are aware that the recent paper [morrison2024invariants] by Morrison-Walker-Wedrich has an independent proof of a part of this structural result (except addenda (3) and (4)), stated in a more general form. They extract slightly different invariants from the filtration structure, which gives somewhat less information than ours in the $\mathfrak{gl}_{2}$ case. Moreover, our invariants generalize Rasmussen $s$-invariants for links and enjoy nice functorial properties.. For the purpose of exposition, in the current section we assume $L$ is the empty link. Let $q\colon\mathcal{S}_{0}^{Lee}(X)\to\mathbb{Z}\cup\{-\infty\}$ denote the quantum filtration function.

The lasagna $s$-invariants of $X$ thus define a smooth invariant $s(X;\bullet):H_{2}(X)\to\mathbb{Z}\cup\{-\infty\}$.

The genus function of a $4$-manifold $X$, defined by

$$g(X,\alpha):=\min\{g(\Sigma)\colon\Sigma\subset X,[\Sigma]=\alpha\},$$

captures subtle information about the smooth topology of $X$. Like the classical $s$-invariant [rasmussen2010khovanov, beliakova2008categorification], which provides a lower bound on the slice genus of links, the lasagna $s$-invariants provide a lower bound on the genus function of $4$-manifolds.

Unlike the classical $s$-invariant, the lasagna $s$-invariants are not directly computable from link diagrams. However, we are able to compute them in some special cases. We indicate one special case where Theorem 1.8 does provide useful information.

The $n$-shake genus of a knot $K$ is defined by $g_{sh}^{n}(K):=g(X_{n}(K);1)$, where $1$ is a generator of $H_{2}(X_{n}(K))\cong\mathbb{Z}$. Clearly, $g_{sh}^{n}(K)\leq g_{4}(K)$, but the inequality may not be strict [akbulut19772, piccirillo2019shake]. Theorem 1.8 specializes to a shake genus bound

$$g_{sh}^{n}(K)\geq\frac{s(X_{n}(K);1)+n}{2}.$$

(1)

Lasagna $s$-invariants also obstruct embeddings between $4$-manifolds in the following sense.

We state a comparison result for Lee skein lasagna modules of $2$-handlebodies, which can sometimes be used as a nonvanishing criterion for Khovanov skein lasagna modules in view of Theorem 1.12.

Since $-5_{2}$ is a positive knot with $s(-5_{2})=2$, by Theorem 1.9(1) we have $g_{sh}^{-1}(-5_{2})=1$. On the other hand, $P(3,-3,-8)$ is slice, thus $g_{sh}^{-1}(P(3,-3,8))=0$. This shows $int\,X_{1}$ and $int\,X_{2}$ are non-diffeomorphic, without actually calculating $\mathcal{S}_{0}^{\bullet}(X_{i})$.

We still need to argue the Khovanov (and indeed Lee) skein lasagna modules of $X_{1},X_{2}$ actually differ. Apply Theorem 1.14(2) to $-5_{2}$, we obtain $s(X_{1};1)=3$,

$$\mathcal{S}_{0,0,q}^{2}(X_{1};1)\otimes\mathbb{Q}=gr_{q}(\mathcal{S}_{0,0}^{Lee}(X_{1};1))=\begin{cases}\mathbb{Q},&q=1,3\\ 0,&\text{otherwise}.\end{cases}$$

Apply Theorem 1.14(2) to the unknot, and then Proposition 1.13 to $P(3,-3,-8)$ and the unknot, we obtain $s(X_{2};1)=1$, and

$$\mathcal{S}_{0,0,q}^{2}(X_{2};1)\otimes\mathbb{Q}\supset gr_{q}(\mathcal{S}_{0,0}^{Lee}(X_{2};1))=\begin{cases}\mathbb{Q},&q=\pm 1\\ 0,&\text{otherwise}.\end{cases}$$

where the first containment follows from Theorem 1.12. In particular, $\mathcal{S}_{0,0,-1}^{2}(X_{1};1)\otimes\mathbb{Q}\neq\mathcal{S}_{0,0,-1}^{2}(X_{2};1)\otimes\mathbb{Q}$.∎

As an example, we note that the recent work by the first author [ren2023lee] has the following implications on the Lee and Khovanov skein lasagna modules for $\overline{\mathbb{CP}^{2}}$.

It is also worth pointing out that Conjecture 6.1 in [ren2023lee] is equivalent to a complete determination of the Khovanov skein lasagna module of $\overline{\mathbb{CP}^{2}}$ (see Conjecture LABEL:conj:-CP2).

If $\Sigma\subset\mathbb{CP}^{2}\backslash(int(B^{4})\sqcup int(B^{4}))$ is an oriented cobordism in $\mathbb{CP}^{2}$ between two oriented links $L_{0},L_{1}$, the last part of Proposition 1.15 enables us to define an induced map

$$Kh(\Sigma)\colon Kh(L_{0})\to Kh(L_{1})$$

between ordinary Khovanov homology of $L_{0}$ and $L_{1}$, at least over a field and up to sign, with bidegree $(0,\chi(\Sigma)-[\Sigma]^{2}+|[\Sigma]|)$. We carry out this construction in more details in Section LABEL:sbsec:Kh_CP2_cob.

Many similarities between skein lasagna modules and other gauge/Floer-theoretic smooth invariants appear. However, a serious constraint for our computation is that we were not able to produce any $4$-manifold with $b_{2}^{+}>0$ that has finite lasagna $s$-invariants or nonvanishing Khovanov skein lasagna module. In fact, if one were able to find a simply-connected closed $4$-manifold $X$ with $b_{2}^{+}(X)>0$ that has $\mathcal{S}_{0}^{2}(X)\neq 0$, then $X\#\overline{\mathbb{CP}^{2}}$ is an exotic closed $4$-manifold, as it has nonvanishing $\mathcal{S}_{0}^{2}$ while its standard counterpart (some $k\mathbb{CP}^{2}\#\ell\overline{\mathbb{CP}^{2}}$) has vanishing $\mathcal{S}_{0}^{2}$.

To pursue this thread, we propose the following conjecture, which seems plausible and desirable.

The part concerning $s(X;1)$ in Conjecture 1.17 implies the $n$-shake genus of $K$ for $n<TB(K)$ (or more generally $n<TB(K^{\prime})$ for some $K^{\prime}$ concordant to $K$) is bounded below by $s(K)/2$, generalizing Theorem 1.9(1). This seems to be the proper straightening of the bound from the Stein adjunction inequality (2).

As will be clear later, Conjecture 1.17 is equivalent to an assertion on the $s$-invariants of cables of $K$ (or some Legendrian link in the $2$-handlebody case) (cf. Proposition 5.4).

In Section 2, we review the theory of Khovanov skein lasagna modules, expanding on a few properties, as well as making some new comments on the skein-theoretic construction. In the short Section 3, we prove Theorem 1.4 by applying a bound on Khovanov homology by Ng [ng2005legendrian], and we give some examples of $4$-manifolds with vanishing Khovanov skein lasagna module.

In Section 4, we develop the theory of Lee skein lasagna modules and lasagna $s$-invariants, which can be thought of as a lasagna generalization of the classical theory by Lee [lee2005endomorphism] and Rasmussen [rasmussen2010khovanov] for links in $S^{3}$. A generalization of Theorem 1.6, in particular the definition of canonical Lee lasagna generators, is presented in Section 4.1. The behavior of canonical Lee lasagna generators under morphisms is given in Theorem 4.5 in Section 4.2. Building on these results, in Section 4.3 we define and examine the lasagna $s$-invariants, generalizing Definition 1.7, Theorem 1.8, and Proposition 1.11.

In Section 5, we make use of a $2$-handlebody formula by Manolescu-Neithalath [manolescu2022skein] to prove nonvanishing results for Khovanov and Lee skein lasagna modules of $2$-handlebodies. In Section 5.1, we prove Theorem 1.12 which shows that finiteness results for the quantum filtration of Lee skein lasagna modules imply nonvanishing results for Khovanov skein lasagna modules. In Section 5.2 we prove a generalization of Proposition 1.13 and in Section 5.3 we show that lasagna $s$-invariants can be expressed in terms of classical $s$-invariants of some cable links. Finally, in the somewhat technical Section 5.4, we prove a vast generalization of the diagrammatic nonvanishing criterion Theorem 1.14.

Finally, in Section 6, we give examples and applications, mostly to the theory and tools developed in Section 4 and Section 5. This includes proofs of Corollary 1.2, Theorem 1.3, Theorem 1.9, and an expanded discussion of Section 1.6. Other discussions include a proof that $s$-invariants of cables of the Conway knot can obstruct its sliceness, and that the Khovanov skein lasagna module over $\mathbb{Q}$ does not detect potential exotica arising from Gluck twists in $S^{4}$.

Most of our developments are independent to proving Theorem 1.1. The readers only interested in Theorem 1.1 may want to look at its proof in Section 1.5 and delve into the corresponding sections. In particular, to distinguish the smooth structures of the pair $X_{1},X_{2}$ in Theorem 1.1, one can read the first half of Section 2, certain statements in Section 4 (mainly Definition 4.6 and Corollary 4.13), Section 5.4, and Section 6.4. The minimal amount of theory needed to actually distinguish their Khovanov skein lasagna modules depends moreover on Section 5.1 and Section 5.2 (but not really on Corollary 4.13 and Section 6.4). Still, the statements in those sections are often much more general than needed for Theorem 1.1, which at times complicate the proofs.

We thank Ian Agol, Daren Chen, Kyle Hayden, Sungkyung Kang, Ciprian Manolescu, Marco Marengon, Anubhav Mukherjee, Ikshu Neithalath, Lisa Piccirillo, Mark Powell, Yikai Teng, Joshua Wang, Paul Wedrich, Hongjian Yang for helpful discussions. This work was supported in part by the Simons grant “new structures in low-dimensional topology” through the workshop on exotic $4$-manifolds held at Stanford University December 16-18, 2023.

Let $X$ be a $4$-manifold and $L\subset\partial X$ be a framed oriented link on its boundary. A skein in $(X,L)$ or in $X$ rel $L$ is a properly embedded framed oriented surface in $X$ with the interiors of finitely many disjoint $4$-balls deleted, whose boundary on $\partial X$ is $L$. That is, a skein in $(X,L)$ can be written as

$$\Sigma\subset X\backslash{\sqcup_{i=1}^{k}int(B_{i})},\text{ with }\partial\Sigma|_{\partial X}=L,$$

(3)

The deleted balls $B_{i}$ are called input balls of the skein $\Sigma$.

We define the category of skeins in $(X,L)$, denoted $\mathcal{C}(X;L)$, as follows. The objects consist of all skeins in $X$ rel $L$. Suppose $\Sigma$ is an object as in (3) and $\Sigma^{\prime}$ is another object with input balls $B_{j}^{\prime}$, $j=1,\cdots,k^{\prime}$. If $\sqcup B_{i}\subset\sqcup B_{j}^{\prime}$ and each $B_{j}^{\prime}$ either coincides with some $B_{i}$ or does not intersect any $B_{i}$ on its boundary, then the set of morphisms from $\Sigma$ to $\Sigma^{\prime}$ consists of pairs of an isotopy class of surfaces⁵⁵5We formally allow some components of $S$ to be curves on the boundaries of those $B_{i}=B_{j}^{\prime}$. $S\subset\sqcup B_{j}^{\prime}\backslash\sqcup\,int(B_{i})$ rel boundary with $S|_{\partial B_{i}}=\Sigma|_{\partial B_{i}}$, $S|_{\partial B_{j}^{\prime}}=\Sigma^{\prime}|_{\partial B_{j}^{\prime}}$, and an isotopy class of isotopies from $S\cup\Sigma^{\prime}$⁶⁶6In order for the surfaces to glue smoothly, one may assume throughout that all embedded surfaces have a standard collar near each boundary, and all isotopies rel boundary respect these collars. to $\Sigma$ rel boundary. We will suppress the isotopy and write $[S]\colon\Sigma\to\Sigma^{\prime}$ for this morphism. If the input balls $B_{i},B_{j}^{\prime}$ do not satisfy the condition above, there is no morphism from $\Sigma$ to $\Sigma^{\prime}$. The identity morphism at $\Sigma$ is given by $[\Sigma|_{\sqcup\partial B_{i}}]$. The composition of two morphisms $[S]\colon\Sigma\to\Sigma^{\prime}$ and $[S^{\prime}]\colon\Sigma^{\prime}\to\Sigma^{\prime\prime}$ is $[S\cup S^{\prime}]$ (together with the composition of isotopies).

Next we introduce an arbitrary (stratified) TQFT of framed oriented links in $S^{3}$, by which we mean a symmetric monoidal functor $Z$ from the category of framed oriented links in $S^{3}$ and link cobordisms in $S^{3}\times I$ up to isotopy rel boundary, to the category of modules over a commutative ring $R$, such that $Z(\emptyset)=R$ and $Z(L)$ is finitely generated for all links $L$. Equivalently, $Z$ is a functorial link invariant in $\mathbb{R}^{3}$ satisfying the conditions in Theorem 2.1 of [morrison2024invariants]. At times we also assume $R$ is a field and $Z$ is involutive, meaning that $Z(-L)\cong Z(L)^{*}$ for a link $L$ and $Z(-\Sigma)\cong Z(\Sigma)^{*}$ for a cobordism $\Sigma\colon L_{0}\to L_{1}$, where $-\Sigma\colon-L_{1}\to-L_{0}$ is the reverse cobordism.

Assign to every object $\Sigma$ of $\mathcal{C}(X;L)$ as in the notations before the $R$-module $\bigotimes_{i=1}^{k}Z(K_{i})$ which we denote by an abuse of notation $Z(\Sigma)$, where each $K_{i}=\partial\Sigma|_{\partial B_{i}}$ is a framed link. Assign to every morphism $[S]\colon\Sigma\to\Sigma^{\prime}$ the morphism $Z([S])\colon Z(\Sigma)\to Z(\Sigma^{\prime})$ induced by the cobordism $S\colon\sqcup K_{i}\to\sqcup K_{j}^{\prime}$. Then $Z$ defines a functor from the category of skeins in $(X,L)$ to the category of $R$-modules.

The skein lasagna module of $(X,L)$ with respect to $Z$, denoted $\mathcal{S}_{0}^{Z}(X;L)$, can be now defined as

$$\mathcal{S}_{0}^{Z}(X;L):=\mathrm{colim}(Z\colon\mathcal{C}(X;L)\to\mathbf{Mod}_{R}).$$

(4)

More explicitly,

$$\mathcal{S}_{0}^{Z}(X;L)=\{(\Sigma,v)\colon\Sigma\subset X\backslash\sqcup_{i=1}^{k}int(B_{i}),\ v\in Z(\Sigma)\}/\sim$$

(5)

where $\sim$ is the equivalence relation generated by linearity in $v$, and $(\Sigma,v)\sim(\Sigma^{\prime},Z([S])(v))$ for $[S]\colon\Sigma\to\Sigma^{\prime}$. A pair $(\Sigma,v)$ (or just $v$ if there’s no confusion) as in (5) is called a lasagna filling of $(X,L)$.

The skein lasagna module $\mathcal{S}_{0}^{Z}(X;L)$ is an invariant of the pair $(X,L)$ up to diffeomorphism. When $L=\emptyset$, we write for short $\mathcal{S}_{0}^{Z}(X)$ for $\mathcal{S}_{0}^{Z}(X;L)$. In this case, since all lasagna fillings and their relations appear in the interior of $X$, $\mathcal{S}_{0}^{Z}(X)$ is an invariant of $int(X)$.

We state some properties of the skein lasagna modules, mostly following Manolescu-Neithalath [manolescu2022skein]. Although their paper dealt with the case when the TQFT $Z$ is the Khovanov-Rozansky $\mathfrak{gl}_{N}$ homology, the proofs remain valid in the more general setup. We remark that the more recent paper [manolescu2023skein] gives a formula for $\mathcal{S}_{0}^{Z}(X;L)$ in terms of a handlebody decomposition of $X$. However, we will not be using this result, so we refer interested readers to their paper.

Every object $\Sigma$ of $\mathcal{C}(X;L)$ represents a homology class in $H_{2}(X,L)$, whose image under the boundary homomorphism $\partial$ in the long exact sequence

$$0\to H_{2}(X)\to H_{2}(X,L)\xrightarrow{\partial}H_{1}(L)\to H_{1}(X)\to\cdots$$

is $[L]$, the fundamental class of $L$. If $[S]\colon\Sigma\to\Sigma^{\prime}$ is a morphism, then $[\Sigma]=[\Sigma^{\prime}]\in H_{2}(X,L)$. This means $\mathcal{S}_{0}^{Z}(X;L)$ admits a grading by $H_{2}^{L}(X):=\partial^{-1}([L])$, which we write as

$$\mathcal{S}_{0}^{Z}(X;L)=\bigoplus_{\alpha\in H_{2}^{L}(X)}\mathcal{S}_{0}^{Z}(X;L;\alpha).$$

Of course, $H_{2}^{L}(X)$ is nonempty if and only if the image of $[L]$ in $H_{1}(X)$ is zero, i.e. $L$ is null-homologous in $X$. From now on this will be assumed, since otherwise $\mathcal{S}_{0}^{Z}(X;L)$ is trivially zero as there is no lasagna filling.

Suppose $(X,L)$ is a pair and $Y\subset X$ is a properly embedded separating oriented $3$-manifold, possibly with transverse intersections with $L$ in some point set $P\subset\partial Y$. Then $Y$ cuts $(X,L)$ into two pairs $(X_{1},T_{1})$ and $(X_{2},T_{2})$, where $\partial X_{1}$ induces the orientation on $Y$ and $\partial X_{2}$ induces the reverse orientation, and $T_{1},T_{2}$ are framed oriented tangles with endpoints $P$. The framing and the orientation of $L$ restrict to a framing and an orientation of $P\subset\partial Y$.

If $T_{0}\subset Y$ is a framed oriented tangle with $\partial T_{0}=P$ in the framed oriented sense, then $T_{1}\cup T_{0}$ is a framed oriented link in $\partial X_{1}$ and $-T_{0}\cup T_{2}$ is one in $\partial X_{2}$. A lasagna filling of $(X_{1},T_{1}\cup T_{0})$ and a lasagna filling of $(X_{2},-T_{0}\cup T_{2})$ glue to a lasagna filling of $(X,L)$ in a bilinear way, inducing a map

$$\mathcal{S}_{0}^{Z}(X_{1};T_{1}\cup T_{0})\otimes\mathcal{S}_{0}^{Z}(X_{2};-T_{0}\cup T_{2})\to\mathcal{S}_{0}^{Z}(X;L)$$

(6)

Write $L_{1}=T_{1}\cup T_{0}$. A framed surface $S\subset X_{2}$ with $\partial S=-T_{0}\cup T_{2}$ is a lasagna filling of $(X_{2};-T_{0}\cup T_{2})$ (without input balls), thus putting its class in the second tensor summand of (6) defines a gluing map

$$\mathcal{S}_{0}^{Z}(X_{2};S)\colon\mathcal{S}_{0}^{Z}(X_{1};L_{1})\to\mathcal{S}_{0}^{Z}(X;L).$$

(7)

The most useful cases of gluing might be when $Y$ is closed, where $P=\emptyset$, and $T_{0},T_{1},T_{2}$ are links. Still more specially, if $X\subset int(X^{\prime})$ is an inclusion of $4$-manifolds, there is an induced map $\mathcal{S}_{0}^{Z}(X)\to\mathcal{S}_{0}^{Z}(X^{\prime})$ by gluing the exterior of $X\subset X^{\prime}$ to $X$. For example, the map in Proposition 2.1 is defined this way.

The skein lasagna module with respect to $Z$ refines the TQFT $Z$ in the following sense.

Finally, we remark that extra structures on the TQFT $Z$ usually induce structures on the skein lasagna module $\mathcal{S}_{0}^{Z}$. For example, suppose $Z$ takes values in $R$-modules with a grading by an abelian group $A$, so that for a link cobordism $S$, $Z(S)$ is homogeneous with degree $\chi(S)a$ for some $a\in A$ independent of $S$. Then the lasagna module $\mathcal{S}_{0}^{Z}(X;L)$ inherits an $A$-grading, once we impose an extra grading shift of $\chi(\Sigma)a$ in the definition of $Z(\Sigma)$ for a skein $\Sigma$. Analogously, a filtration structure on $Z$ induces a filtration structure on $\mathcal{S}_{0}^{Z}$. The isomorphisms or morphisms stated or constructed in this section all respect the extra grading or filtration structure, with a warning that the gluing morphism $\mathcal{S}_{0}^{Z}(X_{2};S)$ in (7) has degree or filtration degree $(\chi(S)-\chi(T_{0}))a$.