98 research outputs found
Floer homology and singular knots
We define Floer homology theories for oriented, singular knots in S^3 and
show that one of these theories can be defined combinatorially for planar
singular knots.Comment: Minor revision
A cube of resolutions for knot Floer homology
We develop a skein exact sequence for knot Floer homology, involving singular
knots. This leads to an explicit, algebraic description of knot Floer homology
in terms of a braid projection of the knot.Comment: 55 pages, 24 figure
Combinatorial Heegaard Floer homology and nice Heegaard diagrams
We consider a stabilized version of hat Heegaard Floer homology of a
3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed
3-manifolds). We give a combinatorial algorithm for constructing this
invariant, starting from a Heegaard decomposition for Y, and give a
combinatorial proof of its invariance properties
Knot Floer homology and the four-ball genus
We use the knot filtration on the Heegaard Floer complex to define an integer
invariant tau(K) for knots. Like the classical signature, this invariant gives
a homomorphism from the knot concordance group to Z. As such, it gives lower
bounds for the slice genus (and hence also the unknotting number) of a knot;
but unlike the signature, tau gives sharp bounds on the four-ball genera of
torus knots. As another illustration, we calculate the invariant for several
ten-crossing knots.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper17.abs.htm
Heegaard Floer homology and alternating knots
In an earlier paper, we introduced a knot invariant for a null-homologous
knot K in an oriented three-manifold Y, which is closely related to the
Heegaard Floer homology of Y. In this paper we investigate some properties of
these knot homology groups for knots in the three-sphere. We give a
combinatorial description for the generators of the chain complex and their
gradings. With the help of this description, we determine the knot homology for
alternating knots, showing that in this special case, it depends only on the
signature and the Alexander polynomial of the knot (generalizing a result of
Rasmussen for two-bridge knots). Applications include new restrictions on the
Alexander polynomial of alternating knots.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper6.abs.htm
Bordered Floer homology and the spectral sequence of a branched double cover I
Given a link in the three-sphere, Z. Szab\'o and the second author
constructed a spectral sequence starting at the Khovanov homology of the link
and converging to the Heegaard Floer homology of its branched double-cover. The
aim of this paper and its sequel is to explicitly calculate this spectral
sequence, using bordered Floer homology. There are two primary ingredients in
this computation: an explicit calculation of filtered bimodules associated to
Dehn twists and a pairing theorem for polygons. In this paper we give the first
ingredient, and so obtain a combinatorial spectral sequence from Khovanov
homology to Heegaard Floer homology; in the sequel we show that this spectral
sequence agrees with the previously known one.Comment: 45 pages, 16 figures. v2: Published versio
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