Instanton Floer homology and the Alexander polynomial

TL;DR

This paper establishes a connection between instanton Floer homology of knots and the Alexander polynomial, showing that the homology's eigenspace Euler characteristics correspond to polynomial coefficients and that it detects fibered knots.

Contribution

It proves that the Euler characteristics of instanton Floer homology eigenspaces match the Alexander polynomial coefficients and demonstrates that instanton homology detects fibered knots.

Findings

Euler characteristics of eigenspaces equal Alexander polynomial coefficients

Instanton homology detects fibered knots

Decomposition of Floer homology into eigenspaces

Abstract

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.