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
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M25
References
Publication
Received: 27 July 2009
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