13 research outputs found
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
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 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
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
Bimodules in bordered Heegaard Floer homology
Bordered Heegaard Floer homology is a three-manifold invariant which
associates to a surface F an algebra A(F) and to a three-manifold Y with
boundary identified with F a module over A(F). In this paper, we establish
naturality properties of this invariant. Changing the diffeomorphism between F
and the boundary of Y tensors the bordered invariant with a suitable bimodule
over A(F). These bimodules give an action of a suitably based mapping class
group on the category of modules over A(F). The Hochschild homology of such a
bimodule is identified with the knot Floer homology of the associated open book
decomposition. In the course of establishing these results, we also calculate
the homology of A(F). We also prove a duality theorem relating the two versions
of the 3-manifold invariant. Finally, in the case of a genus one surface, we
calculate the mapping class group action explicitly. This completes the
description of bordered Heegaard Floer homology for knot complements in terms
of the knot Floer homology.Comment: 153 pages, 29 figures; v4: Address referee comment
Grid homology for knots and links
Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the a