30 research outputs found
Combinatorial Description of Knot Floer Homology of Cyclic Branched Covers
We introduce a simple combinatorial method for computing all versions of the
knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its
double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we
obtain looks like a twisted version of the toroidal grid diagrams recently
introduced by Manolescu, Ozsvath, and Sarkar. We conclude with a discussion of
how one might obtain nice Heegaard diagrams for cyclic branched covers of more
general knots.Comment: 20 pages, 14 figures; Minor expositional improvements, typos
corrected throughout (most seriously, the x,y coordinates used in discussion
of intersection points beginning page 13 of previous version were
incorrectly--but consistently--flipped
Knot Floer homology in cyclic branched covers
In this paper, we introduce a sequence of invariants of a knot K in S^3: the
knot Floer homology groups of the preimage of K in the m-fold cyclic branched
cover over K. We exhibit the knot Floer homology in the m-fold branched cover
as the categorification of a multiple of the Turaev torsion in the case where
the m-fold branched cover is a rational homology sphere. In addition, when K is
a 2-bridge knot, we prove that the knot Floer homology of the lifted knot in a
particular Spin^c structure in the branched double cover matches the knot Floer
homology of the original knot K in S^3. We conclude with a calculation
involving two knots with identical knot Floer homology in S^3 for which the
knot Floer homology groups in the double branched cover differ as Z_2-graded
groups.Comment: This is the version published by Algebraic & Geometric Topology on 25
September 200
Sutured Khovanov homology distinguishes braids from other tangles
We show that the sutured Khovanov homology of a balanced tangle in the
product sutured manifold D x I has rank 1 if and only if the tangle is isotopic
to a braid.Comment: 9 pages, 1 figure, Definition of sutured annular Khovanov homology in
Section 2.1 has significant text overlap with arXiv:1212.2222; Version 2
incorporates referee comments. This is the version accepted for publication
in Mathematical Research Letter
On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology
Ozsvath and Szabo have established an algebraic relationship, in the form of
a spectral sequence, between the reduced Khovanov homology of (the mirror of) a
link L in S^3 and the Heegaard Floer homology of its double-branched cover.
This relationship has since been recast by the authors as a specific instance
of a broader connection between Khovanov- and Heegaard Floer-type homology
theories, using a version of Heegaard Floer homology for sutured manifolds
developed by Juhasz. In the present work we prove the naturality of the
spectral sequence under certain elementary TQFT operations, using a
generalization of Juhasz's surface decomposition theorem valid for decomposing
surfaces geometrically disjoint from an imbedded framed link.Comment: 36 pages, 13 figure
On Gradings in Khovanov homology and sutured Floer homology
We discuss generalizations of Ozsvath-Szabo's spectral sequence relating
Khovanov homology and Heegaard Floer homology, focusing attention on an
explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in
the two theories. These two gradings have simple representation-theoretic
(resp., geometric) interpretations, which we also review.Comment: 17 pages, 5 figures, to be submitted to Proceedings of Jaco's 70th
Birthday Conference, 201
Grid Diagrams and Legendrian Lens Space Links
Grid diagrams encode useful geometric information about knots in S^3. In
particular, they can be used to combinatorially define the knot Floer homology
of a knot K in S^3, and they have a straightforward connection to Legendrian
representatives of K in (S^3, \xi_\st), where \xi_\st is the standard, tight
contact structure. The definition of a grid diagram was extended to include a
description for links in all lens spaces, resulting in a combinatorial
description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In
the present article, we explore the connection between lens space grid diagrams
and the contact topology of a lens space. Our hope is that an understanding of
grid diagrams from this point of view will lead to new approaches to the Berge
conjecture, which claims to classify all knots in S^3 upon which surgery yields
a lens space.Comment: 27 pages, 20 figure