[Paper Review] Khovanov homology for links in $\#^r(S^2 imes S^1)$

Michael Willis|arXiv (Cornell University)|Dec 17, 2018

Geometric and Algebraic Topology20 references4 citations

TL;DR

This paper extends Khovanov homology to links in the connected sum of r copies of S²×S¹, denoted Mr, by adapting Rozansky's approach using infinite full twists and Bar-Natan's formal cobordism framework. The construction yields a Z-graded homology theory categorifying both the skein module S(Mr) and WRT invariants, with invariance under isotopy established via handle-slide and Reidemeister move compatibility.

ABSTRACT

We revisit Rozansky's construction of Khovanov homology for links in $S^2 imes S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2 imes S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be used to recover WRT invariants at certain roots of unity, and also recovers the evaluation of $L$ in the skein module $\mathcal{S}(M^r)$ of Hoste and Przytycki when $L$ is null-homologous in $M^r$. The construction also allows for a clear path towards defining a Lee's homology $Kh'(L)$ and associated $s$-invariant for such $L$, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in $S^3$ and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

Motivation & Objective

• To generalize Khovanov homology from S³ to the 3-manifold Mr = #r(S²×S¹), which is not a homology sphere and presents topological challenges.
• To provide a homology theory that categorifies both the skein module S(Mr) of Hoste and Przytycki and Witten-Reshetikhin-Turaev (WRT) invariants at specific roots of unity.
• To establish invariance under isotopy in Mr, including handle-slides and strand movements through attaching spheres, by leveraging properties of full twists and truncated complexes.
• To lay the foundation for defining Lee-type homology and an s-invariant for links in Mr, as well as a potential stable homotopy type for knotifications in future work.

Proposed method

• Represent links in Mr via tangle diagrams with endpoints on r disjoint 2-spheres (belt spheres), using dashed lines to denote surgery spheres.
• Construct a lift to S³ by replacing each dashed line with a torus braid of ki full right-handed twists, forming a diagram L(⃗k) with ki → ∞.
• Use Bar-Natan’s universal formal cobordism category to define a limiting chain complex in the Temperley-Lieb category as ki → ∞.
• Apply Khovanov’s functor to this limiting complex to obtain an infinite chain complex of graded abelian groups.
• Define Kh(L) as the (graded) homology of this complex, which is invariant under isotopy in Mr.
• Prove invariance under handle-slides and strand-pushing moves by showing that the truncated complex C∗>a(ki) behaves consistently under dot-sliding and sign cancellation via even crossing counts.

Experimental results

Research questions

• RQ1 Can Khovanov homology be extended to links in #r(S²×S¹) in a way that categorifies both the skein module S(Mr) and WRT invariants?
• RQ2 How can the infinite twist limit construction be made rigorous and invariant under isotopy in Mr, including non-trivial handle-slides?
• RQ3 What role do truncated complexes C∗>a(ki) and their dot-sliding properties play in proving invariance under strand-pushing and handle-slides?
• RQ4 Can the construction be adapted to define a Lee-type homology and associated s-invariant for links in Mr?
• RQ5 Is there a pathway to lifting this homology theory to a stable homotopy type for knotified links in Mr, as in Lipshitz-Sarkar’s work?

Key findings

• Khovanov homology Kh(L) is well-defined for any link L in Mr with even geometric intersection numbers ni with the belt spheres S²i, and is invariant under ambient isotopy in Mr up to overall grading shifts.
• The graded Euler characteristic of Kh(L) recovers the WRT invariant of L in Mr at certain roots of unity, providing a categorification of quantum invariants.
• For null-homologous links in Mr, the grading shifts vanish, and Kh(L) recovers the evaluation of L in the skein module S(Mr), as defined by Hoste and Przytycki.
• The construction is invariant under handle-slides and strand-pushing through attaching spheres, established via dot-sliding in truncated complexes with even crossing counts ensuring sign cancellation.
• The limiting complex for the torus braid Fki ni as ki → ∞ can be expressed without strands passing from top to bottom, enabling consistent strand-pulling through gaps.
• The method avoids the derived category and Hochschild homology used in Rozansky’s original S²×S¹ construction, instead using Bar-Natan’s formal cobordism framework throughout, enabling clearer extension to higher genus manifolds.