69 research outputs found
Knot Floer homology detects fibred knots
Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred
knots in . We will prove this conjecture for null-homologous knots in
arbitrary closed 3--manifolds. Namely, if is a knot in a closed 3--manifold
, is irreducible, and is monic, then is fibred.
The proof relies on previous works due to Gabai, Ozsv\'ath--Szab\'o, Ghiggini
and the author. A corollary is that if a knot in admits a lens space
surgery, then the knot is fibred.Comment: version 4: incorporates referee's suggestions, to appear in
Inventiones Mathematica
Notes on bordered Floer homology
This is a survey of bordered Heegaard Floer homology, an extension of the
Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is
placed on how bordered Heegaard Floer homology can be used for computations.Comment: 73 pages, 29 figures. Based on lectures at the Contact and Symplectic
Topology Summer School in Budapest, July 2012. v2: Fixed many small typo
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
Legendrian knots, transverse knots and combinatorial Floer homology
Using the combinatorial approach to knot Floer homology, we define an
invariant for Legendrian knots in the three-sphere, which takes values in link
Floer homology. This invariant can be used to also construct an invariant of
transverse knots.Comment: 27 pages, 13 figures; v2: Expand and correct discussion of link
Heegaard Floer homology as morphism spaces
In this paper we prove another pairing theorem for bordered Floer homology.
Unlike the original pairing theorem, this one is stated in terms of
homomorphisms, not tensor products. The present formulation is closer in spirit
to the usual TQFT framework, and allows a more direct comparison with
Fukaya-categorical constructions. The result also leads to various dualities in
bordered Floer homology.Comment: 57 pages, 14 figures; v2: many updates, including changing
orientation conventions, which changed the signs in many theorem
Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is
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 is to explicitly calculate this spectral sequence in terms of
bordered Floer homology. There are two primary ingredients in this computation:
an explicit calculation of bimodules associated to Dehn twists, and a general
pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on
computing the bimodules; this part focuses on the pairing theorem for polygons,
in order to prove that the spectral sequence constructed in the previous part
agrees with the one constructed by Ozsv\'ath and Szab\'o.Comment: 85 pages, 19 figures, v3: Version to appear in Journal of Topolog
Computing HF^ by factoring mapping classes
Bordered Heegaard Floer homology is an invariant for three-manifolds with
boundary. In particular, this invariant associates to a handle decomposition of
a surface F a differential graded algebra, and to an arc slide between two
handle decompositions, a bimodule over the two algebras. In this paper, we
describe these bimodules for arc slides explicitly, and then use them to give a
combinatorial description of HF^ of a closed three-manifold, as well as the
bordered Floer homology of any 3-manifold with boundary.Comment: 106 pages, 46 figure
Localization for Involutions in Floer Cohomology
We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.National Science Foundation (U.S.) (grant DMS-0405516)National Science Foundation (U.S.) (grant DMS-065260)European Research Council (grant ERC-2007-StG-205349
Ricci flow for homogeneous compact models of the universe
Using quaternions, we give a concise derivation of the Ricci tensor for
homogeneous spaces with topology of the 3-dimensional sphere. We derive
explicit and numerical solutions for the Ricci flow PDE and discuss their
properties. In the collapse (or expansion) of these models, the interplay of
the various components of the Ricci tensor are studied. We dedicate this paper
to honor the work of Josh Goldberg.Comment: 18 pages, 2 figure
Filtrations on the knot contact homology of transverse knots
We construct a new invariant of transverse links in the standard contact
structure on R^3. This invariant is a doubly filtered version of the knot
contact homology differential graded algebra (DGA) of the link. Here the knot
contact homology of a link in R^3 is the Legendrian contact homology DGA of its
conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations
are constructed by counting intersections of the holomorphic disks of the DGA
differential with two conormal lifts of the contact structure. We also present
a combinatorial formula for the filtered DGA in terms of braid representatives
of transverse links and apply it to show that the new invariant is independent
of previously known invariants of transverse links.Comment: 23 pages, v2: minor corrections suggested by refere
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