Link homology and equivariant gauge theory

TL;DR

This paper connects link homology and equivariant gauge theory, providing topological insights and explicit computations for singular instanton Floer homology of knots and links, especially relating gradings to knot signatures.

Contribution

It introduces a topological method to determine gradings in singular instanton Floer homology and computes generators for various knots and links using equivariant gauge theory.

Findings

Graded the special generator by knot signature mod 4

Explicitly computed generators for two-bridge, torus, and Montesinos knots

Extended computations to certain two-component links

Abstract

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links.