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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.