146 research outputs found
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
Filtrations on instanton homology
In earlier work of the authors, the Khovanov complex of a knot or link
appeared as the first page in a spectral sequence abutting to the instanton
homology. The quantum and (co)homological gradings on Khovanov homology do not
survive as gradings, but we show that they survive as filtrations.Comment: 40 pages, 4 figures. Revised version, with corrected typos and
extended introductio
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