The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

TL;DR

This paper constructs a functorial invariant of 2-stranded tangles using Fukaya categories and traceless SU(2) character varieties, connecting symplectic geometry with Khovanov homology and tangle invariants.

Contribution

It introduces a new geometric approach to tangle invariants via Fukaya categories and character varieties, providing a functorial framework that relates to Khovanov homology.

Findings

Invariant is functorial under isotopy.

Hom set matches reduced Khovanov chain complex.

Provides a geometric interpretation of tangle invariants.

Abstract

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.