709 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
Index theory of the de Rham complex on manifolds with periodic ends
We study the de Rham complex on a smooth manifold with a periodic end modeled
on an infinite cyclic cover X' \to X. The completion of this complex in
exponentially weighted L^2-norms is Fredholm for all but finitely many
exceptional weights determined by the eigenvalues of the covering translation
map H_*(X') \to H_*(X'). We calculate the index of this weighted de Rham
complex for all weights away from the exceptional ones
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
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