We describe a weighted $A_infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fuka
ya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two differe
Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue
of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.
In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading 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.
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this re
quires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and th
en using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams.
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle int
roduction to the subject, and does not itself have immediate topological applications.
