Instanton Floer homology and the Alexander polynomial

Kronheimer, Peter; Mrowka, Tomasz

doi:10.2140/agt.2010.10.1715

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Instanton Floer homology and the Alexander polynomial

Peter Kronheimer and Tom Mrowka

Algebraic & Geometric Topology 10 (2010) 1715–1738

DOI: 10.2140/agt.2010.10.1715

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

Volume 10, issue 3 (2010)