An equivalence between combinatorial tangle floer and contact categories

We prove an equivalence between the category underlying combinatorial tangle Floer homology and the contact category by building on the prior work of Lipshitz, Ozsváth, and Thurston and later Zhan. In his 2015 paper "Formal Contact Categories", Cooper establishes a relationship between the categories associated to oriented surfaces by Heegaard Floer theory and embedded contact theory. In this thesis, we examine a special case of his general argument to show an equivalence between the categories discussed by Petkova and Vértesi and those discussed by Tian. To do this, we construct two bimodules associated to the transformations between the underlying structure of combinatorial tangle Floer homology and the contact category. We take the tensor product of these bimodules and show that the product is equivalent to the identity, inducing an isomorphism between the categories of interest.