Tangle Floer homology and cobordisms between tangles

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 $H{F}^M$ 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 $H{F}^M(T)$ 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.

66 pages, Corrected Typos, Section 8.3 is moved to a new paper: arXiv:1810.05125

References in corpus (3)

• Link cobordisms and functoriality in link Floer homology
• The Bar-Natan homology and unknotting number
• Unoriented Cobordism Maps on Link Floer Homology

Cited by in corpus (4)

• Link cobordisms and functoriality in link Floer homology
• Knot cobordisms, bridge index, and torsion in Floer homology
• Knot Floer homology and the unknotting number
• Peculiar modules for 4-ended tangles