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Register a new userNon-orientable link cobordisms and torsion order in Floer homologies
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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.
We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.
In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a null-homologous link in a three-manifold.
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish.
Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.
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