The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties

TL;DR

This paper links the Bar-Natan skein module of a solid torus to the homology of (n,n) Springer varieties, providing new algebraic formulas and deepening the understanding of their topological and algebraic structures.

Contribution

It establishes an isomorphism between the skein module and Springer variety homology, and derives a formula for comultiplication in this context.

Findings

Isomorphism between skein module and Springer homology

Explicit formula for comultiplication in the skein module

Extension of Khovanov's work to new topological settings

Abstract

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.