Volume 10, issue 3 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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