Tangle Floer homology and cobordisms between tangles

TL;DR

This paper introduces a generalized tangle Floer homology framework that associates algebraic invariants to tangles and their cobordisms, extending sutured Floer homology and 4-manifold invariants.

Contribution

It develops a new category of A-tangles and cobordisms, defining a functorial tangle Floer homology that generalizes previous invariants and applies to decorated link cobordisms.

Findings

Defines A-tangles with colorings and SpinC structures.

Constructs a functor from A-tangles to A-modules, extending sutured Floer homology.

Generalizes 4-manifold invariants of Ozsvath and Szabo.

Abstract

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed SpinC structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an affine set of SpinC structures. Associated with every A-module M we construct a functor $HF^M$ from A-Tangles to A-Modules, called the tangle Floer homology functor, where A-Modules denotes the the category of A-modules and A-homomorphisms between them. Moreover, for any A-tangle T the A-module $HF^M(T)$ is the extension of sutured Floer homology defined in an earlier work of the authors. In particular, this construction generalizes the 4-manifold invariants of Ozsvath and Szabo. Moreover, applying the above machinery to decorated cobordisms between links, we get functorial maps on link Floer homology.