Instanton Floer homology and the Alexander polynomial

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.Comment: 25 pages, 6 figures. Revised version, correcting errors concerning mod 2 gradings in the skein sequenc

Gauge theory and Rasmussen's invariant

A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpart of s(K) is not additive for connected sums.Comment: This version bundles the original submission with a 1-page corrigendum, indicating the error. The new version of the corrigendum points out that the invariant is not additive for connected sums. 23 pages, 3 figure

Exact Triangles for SO(3) Instanton Homology of Webs

The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. From the octahedral diagram, one can derive equivalent reformulations of the authors' conjecture that, for planar webs, the rank of the instanton homology is equal to the number of Tait colorings.National Science Foundation (U.S.) (Grant DMS-0805841)National Science Foundation (U.S.) (Grant DMS-1406348

PU(2) monopoles and a conjecture of Marino, Moore, and Peradze

In this article we show that some of the recent results of Marino, Moore, and Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten simple type) are of `superconformal simple type' -- can be understood using a simple mathematical argument via the PU(2)-monopole cobordism of Pidstrigach and Tyurin (dg-ga/9507004) and results of the first and third authors (dg-ga/9712005, dg-ga/9709022).Comment: 13 pages, 1 figure. Improved exposition, typographical slips corrected, figure and references added. Minor correction on page 2. To appear in Mathematical Research Letter

Witten's conjecture and Property P

Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a non-trivial homomorphism to the group SO(3). In particular, Y cannot be a homotopy-sphere.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper7.abs.html Version 5: links correcte