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Tangle Floer homology and cobordisms between tangles
Journal of Topology  (IF1.582),  Pub Date : 2020-09-24, DOI: 10.1112/topo.12168
Akram Alishahi, Eaman Eftekhary

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 $Z / 2 Z$, 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 $Spin c$ structures, and $A$‐cobordisms as morphisms. An $A$‐cobordism is a cobordism with a compatible $A$‐coloring and an isotropic set of $Spin c$ structures. Associated with every $A$‐module $M$ we construct a functor
$HF M : A - Tangles ⟶ A - Modules ,$
called the tangle Floer homology functor, where $A - Modules$ denotes 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.