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We use Kauffmans bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.
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We define stable homotopy refinements of Khovanovs arc algebras and tangle invariants.
We study generalizations of a classical link invariant -- the multivariable Alexander polynomial -- to tangles. The starting point is Archibalds tMVA invariant for virtual tangles which lives in the setting of circuit algebras, and whose target space has dimension that is exponential in the number of strands. Using the Hodge star map and restricting to tangles without closed components, we define a reduction of the tMVA to an invariant rMVA which is valued in matrices with Laurent polynomial entries, and so has a much more compact target space. We show the rMVA has the structure of a metamonoid morphism and is further equivalent to a tangle invariant defined by Bar-Natan. This invariant also reduces to the Gassner representation on braids and has a partially defined trace operation for closing open strands of a tangle.
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 show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D x I has rank 1 if and only if the tangle is isotopic to a braid.
Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define the Kauffman bracket ideal of T to be the ideal I_T generated by the reduced Kauffman bracket polynomials of all closures of T. If this ideal is non-trivial, then T does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any (S^1 x D^2, 2)-tangle, also called a genus-1 tangle, and give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal. Furthermore, we show that if a single-component genus-1 tangle S can be obtained as the partial closure of a (B^3, 4)-tangle T, then I_T = I_S.
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