Homological Knot Invariants from Mirror Symmetry

TL;DR

This paper demonstrates how homological mirror symmetry can produce new, computable knot invariants that categorify the Jones polynomial, solving the knot categorification problem for various Lie algebras.

Contribution

It introduces a new family of mirror pairs where homological mirror symmetry yields meaningful knot invariants, advancing the understanding of knot homology.

Findings

Explicit computation of invariants for simple Lie algebras

Extension to certain Lie superalgebras

Resolution of the knot categorification problem

Abstract

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning -- what are they homologies of? Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand. I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.