The Lee $s$-filtration as a link concordance invariant: strength relative to $d_L$, effective computation, and relation to link Floer homology (Pardon's open problems)

Pardon (2012) generalized Rasmussen's $s$-invariant from knots to links by pulling back the quantum filtration on Lee homology to $\mathbb{O}(L)$, the $2^{|L|}$-dimensional vector space spanned by the orientations of the link, made into a functor of link cobordisms; its numerical shadow $d_L$, recording the associated-graded dimensions, is a smooth link concordance invariant. The paper left open four issues: (1) whether the full filtration (with its cobordism maps) is strictly stronger than $d_L$, with no distinguishing examples known; (2) a conjectural closed formula for $d_{T_{n,n}}$ verified only for $n\le6$; (3) the general structure and effective computation of the filtration for arbitrary links; and (4) the expected Floer-theoretic counterpart, a link generalization of the Ozsváth-Szabó $\tau$-invariant with properties parallel to the $s$-filtration on $\mathbb{O}(L)$. Item (2) is now resolved — Ren (2024) determined the complete quantum filtration structure of the Lee homology of all torus links, confirming Pardon's $T_{n,n}$ conjecture and, independently, the Manolescu-Marengon-Sarkar-Willis adjunction conjecture — while Pardon's sibling $(2,0)$-cabling conjecture (his formula (5-1), verified for odd $p\le11$) remains open. The Floer side of item (4) was realized by Cavallo (2018), who constructed the link invariant $\tau(L)$ (with a dimension function $T_L$ parallel to $d_L$) from filtered link grid homology, and Schuetz (2025) gave an effective algorithm for the orientation-wise Rasmussen invariant $s(L,o)$, showing that $d_L$-level spectral-sequence data determines $s(L,o)$ only up to two-sided bounds. However, no links distinguishing the full $s$-filtration from $d_L$ have been found, no equivalence theorem is known, the full filtration is not effectively computable for general links, and the Lee-type and Floer-type link concordance theories have not been compared. The surviving core — strength, the cabling conjecture, general computability, and the relation to Floer homology of the link $s$-filtration — remains open.

Background

Khovanov homology is a combinatorially defined invariant of oriented links that categorifies the Jones polynomial: to a link diagram it associates a bigraded chain complex, graded by homological degree $h$ and quantum degree $q$. Eun Soo Lee introduced a deformation of this complex over $\mathbb{Q}$ whose homology, now called Lee homology $Kh^{\mathrm{Lee}}(L)$, is the target of a spectral sequence starting from Khovanov homology and is remarkably small: Lee proved that its dimension is $2^{|L|}$, with a basis naturally indexed by the orientations of the link $L$. Throughout, links and concordance are meant in the smooth category: two links are concordant if they cobound a disjoint union of smoothly embedded annuli in $S^3\times[0,1]$, and the slice genus $g_4(L)$ is the minimal genus of a smoothly embedded oriented surface in $B^4$ with boundary $L$.

For a knot $K$, Rasmussen extracted from the quantum filtration on the two-dimensional Lee homology his invariant $s(K)$: the filtration is supported on two adjacent levels $s(K)\pm1$, which defines the even integer $s(K)$. The invariant $s(K)$ is a homomorphism on the smooth knot concordance group, satisfies $|s(K)|\le 2g_4(K)$, and Rasmussen computed $s(T_{p,q})=(p-1)(q-1)$ for the torus knots $T_{p,q}$ (the knots wrapping a curve of slope $(p,q)$ around the standard torus), giving the first proof of the Milnor conjecture $g_4(T_{p,q})=\tfrac{1}{2}(p-1)(q-1)$ that avoids gauge theory.

For a link, Lee homology has dimension $2^{|L|}$ rather than 2, so the naive generalization of $s$ is not a single number, and cobordisms that merge components behave differently from component-preserving ones. Beliakova and Wehrli defined an integer $s(L,o)$ for an oriented link by restricting the quantum filtration to the two-dimensional subspace spanned by a chosen orientation $o$ and its reverse; this suits cobordisms that may merge components but discards information. Pardon then made the entire structure into the invariant. Let $\mathbb{O}(L)$ be the $\mathbb{Q}$-vector space with basis the orientations of $L$, identified with $Kh^{\mathrm{Lee}}(L)$ via Lee's theorem; $\mathbb{O}$ is a functor of link cobordisms, each orientable cobordism $\Sigma$ inducing a map (defined up to sign) that is filtered of degree $\chi(\Sigma)$. The pullback of the quantum filtration (the $s$-filtration) to $\mathbb{O}(L)$ is the natural link generalization of Rasmussen's invariant, and the associated numerical invariant is $d_L:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}_{\ge0}$, where $d_L(h,s)$ is the dimension of the $s$-th associated graded piece of the $h$-graded part of Lee homology. Pardon proved that $d_L$ is a link concordance invariant with properties parallel to the knot case: convolution behavior under disjoint union, covariance under orientation change, and genus bounds for component-preserving cobordisms — for example, a component-preserving orientable cobordism from a Khovanov-thin link (one whose rational Khovanov homology is concentrated on two adjacent diagonals, as for alternating links and their quasi-alternating generalizations — links obtained from the unknot by iterated resolution at crossings whose two branches are themselves quasi-alternating) to a split link with $m$ components has genus at least $\lfloor m/2\rfloor$, so quasi-alternating links are not concordant to split links.

The strength hierarchy among these invariants is only partly understood. Examples show that the function $o\mapsto s(L,o)$ carries less information than $d_L$, but whether $d_L$ determines it in general is not known, and no pair of links with equal $d_L$ but different $s(L,o)$ is known. Pardon expected $d_L$ in turn to be weaker than the full filtered functor: the filtration remembers which linear combinations of orientations lie at which levels and how cobordisms act, while $d_L$ records only dimensions. No examples proving this were known — the first open issue. Second, for the torus links $T_{n,n}$ (the $n$-component closures of the $(n,n)$ torus braid, all components oriented coherently), Pardon computed $d_{T_{n,n}}$ for $n\le6$, and the referee observed a closed pattern in terms of certain $q$-deformed binomial-type polynomials $P_{n,k}$, conjectured for all $n$; a sibling conjecture concerns the $(2,0)$-cablings $L_p$ of the torus knots $T_{2,p}$ (two zero-framed parallel copies, so the linking number vanishes), verified for odd $p\le11$. Third, beyond cases where the Khovanov-to-Lee spectral sequence degenerates, no effective method was known to compute the actual filtration on $\mathbb{O}(L)$: existing programs compute (deformed) Khovanov homology, from which one reads off $d_L$-level data but not the placement of the orientation classes in the filtration. Fourth, on the Floer side, knot Floer homology — a Heegaard-diagram-based knot invariant introduced by Ozsváth and Szabó — yields another slice-genus-bounding concordance invariant $\tau(K)$; for knots, $s$ and $2\tau$ agree on many well-studied classes (alternating, torus, two-bridge knots) but differ in general, with Livingston producing even topologically slice knots with $s\ne 2\tau$. Pardon expected an analogous invariant of links to be definable from link Floer homology, with properties parallel to the $s$-filtration on $\mathbb{O}(L)$; he deliberately proved his basic filtration properties by soft functorial arguments that would apply verbatim to such a Floer-theoretic analog.

Problem Statement

This problem concerns the cluster of issues Pardon (2012) left open about the $s$-filtration on Lee homology as a link concordance invariant. The objective is to determine the strength, structure, computability, and Floer-theoretic counterpart of this invariant, as laid out in the four issues below.

(1) Strength. Decide whether the full $s$-filtration on $\mathbb{O}(L)$ — the filtered vector space together with its cobordism-induced maps, i.e. the data of which linear combinations of orientations lie at which filtration levels and how cobordisms act — is strictly stronger than the numerical invariant $d_L$ as an invariant of smooth link concordance. Either exhibit two links $L_1,L_2$ with $d_{L_1}=d_{L_2}$ (as functions $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}_{\ge0}$) that are nevertheless distinguished by the filtration (through the level placement of the orientation classes or through the cobordism-induced maps) — or prove that the concordance-relevant content of the filtration is determined by $d_L$. Open.

(2) Structure on natural families. Determine $d_L$, and ideally the full filtration, for natural infinite families of links. The flagship case — the conjectured closed formula for $d_{T_{n,n}}$ (Pardon's formula (5-8)), verified for $n\le6$ — has been proved for all $n$ by Ren (2024) as part of a complete determination of the Lee filtration of all torus links, so this item is resolved for torus links. Remaining in this spirit: Pardon's sibling conjecture (his formula (5-1)) for the $(2,0)$-cablings $L_p$ of $T_{2,p}$ (two zero-framed parallel copies, so the linking matrix vanishes) — asserting that, for every odd $p\ge1$,

$\sum_{i,j} d_{L_p}(i,j)\, t^i q^j = 1+q^2+q^{2p-4}+q^{2p-2}$

which Pardon verified for odd $p\le11$ and proved there only up to a two-fold ambiguity. This conjecture is the concrete surviving demand of item (2); closed-form determinations for further natural families (cables, closures of other braids, weaves) are desirable extensions but are not required.

(3) Computation. Develop an effective method to compute the $s$-filtration on $\mathbb{O}(L)$ for arbitrary links — the full filtration, not merely the orientation-wise values $s(L,o)$, which Schuetz's algorithm now computes — beyond the special cases in which the Khovanov-to-Lee spectral sequence degenerates. Partial progress exists (Ren for torus links; Schuetz's algorithm for $s(L,o)$ of general links), but no method computes the full filtration for general links. Open.

(4) Floer counterpart and comparison. Pardon expected an invariant of links similar to $d_L$ to be definable from link Floer homology as a generalization of the Ozsváth-Szabó $\tau$-invariant, with properties parallel to the $s$-filtration on $\mathbb{O}(L)$. That expectation has been realized: Cavallo's link generalization $\tau(L)$, together with the dimension function $T_L$, is constructed from a filtered version of link grid homology (the combinatorial model of link Floer homology) and satisfies the predicted parallel concordance properties. What remains open is the relationship between the two theories: no inequality, class of links on which the Lee-filtration invariants ($d_L$, the filtration on $\mathbb{O}(L)$, or the values $s(L,o)$) and the Floer invariants ($\tau(L)$, $T_L$) agree, or structural correspondence between them is known — unlike the knot level, where the (still only partly understood) comparison of $s$ and $2\tau$ connects the two theories. Determining how the Lee-type and Floer-type link concordance theories relate is open.

Items (1), the surviving cabling conjecture (5-1) in (2), (3) in general, and the comparison in (4) constitute the surviving open core after Ren's resolution of the torus-link part of (2).

References

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