Instanton Floer homology and the Alexander polynomial

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.