Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus

TL;DR

This paper proposes a symplectic geometric approach to constructing reduced Khovanov homology for links in lens spaces, utilizing a partly conjectural Fukaya category of the traceless $SU(2)$ character variety of a twice-punctured torus.

Contribution

It introduces a new method to define Khovanov homology for links in lens spaces via Fukaya categories and symplectic geometry, extending previous work from $S^3$ to more general 3-manifolds.

Findings

Reproduces reduced Khovanov homology for links in $S^3$

Constructs link invariants in $S^2 imes S^1$

Suggests the cohomology may be a link invariant in lens spaces

Abstract

We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,\delta)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,\delta)$ is an isotopy invariant of the tangle diagram. We use $(X,\delta)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.