Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. This paper shows that the knot Floer homology of this lift, with mod 2 coefficients, can be computed from a spectral sequence starting at a type of Khovanov homology already described by Asaeda, Przytycki, and Sikora. We extend the known results about this type of Khovanov homology, and use it to provide a very simple explanation of the case when L is alternating for the obvious projection.Comment: Due to an error in section 8, the results on the contact element have been weakene

We describe the first part of a gluing theory for the bigraded Khovanov homology with Z-coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The construction is modeled off bordered Heegaard-Floer homology, but uses only combinatorial/diagrammatic methods 1 arXiv:1304.0463v3 [math.GT]

The author learned the Heegaard-Floer idea from John Baldwin while at the Mathematical Sciences Research Institute for the program on Homology theories of knots and links in the spring of 2010. While at MSRI, he stumbled on to the constructions in this paper while trying to understand what he was being told, completing the proof of invariance in fall of 2010. John Baldwin and Adam Levine have used this idea, in conjunction with a construction of C. Manolescu, to describe Ozsváth and Szabó's knot Floer homology using spanning trees of a link diagram, [1]. The author would like to thank John Baldwin for those conversations, as well as P. Ozsváth and Z. Szabó for the great idea. He would also like to thank Liam Watson, Matt Hedden, and Tom Mark for listening as he

Abstract. Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle which complements the type D structure previously defined by the author. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. The homotopy type of the type A structure is a tangle invariant, and homotopy equivalences of the type A structure result in chain homotopy equivalences on the Khovanov chain complex. We can use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. This approach adds to the literature even in the case of a connect sum, where the techniques here will allow an exact computation of Khovanov homology from the stuctures for two tangles coming from the summands. Several examples are included, showing in particular how we can compute the correct torsion summands for the Khovanov homology of the connect sum. A lengthy appendix is devoted to establishing the theory of these structures over a characterstic zero ring.

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