#### Volume 10, issue 3 (2010)

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Instanton Floer homology and the Alexander polynomial

### Peter Kronheimer and Tom Mrowka

Algebraic & Geometric Topology 10 (2010) 1715–1738
##### 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.

##### Keywords
knot, Alexander polynomial, instanton, Floer homology, Yang–Mills
Primary: 57R58
Secondary: 57M25
##### Publication
Revised: 2 June 2010
Accepted: 13 June 2010
Published: 11 August 2010
##### Authors
 Peter Kronheimer Department of Mathematics Harvard University Cambridge MA 02138 Tom Mrowka Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139