Chord diagrams and contact-topological quantum field theory
by Dan Matthews
Abstract: We consider the topological quantum field theory properties of sutured Floer homology, as introduced by Honda--Kazez--Matic. In the ``dimensionally reduced" case of product manifolds, the computation of sutured Floer homology and contact elements reduces to that for solid tori with longitudinal sutures. The SFH of such manifolds forms a ``categorification of Pascal's triangle", and contact structures correspond bijectively to chord diagrams, or sets of disjont properly embedded arcs in the disc; contact elements form distinguished subsets of $SFH$ of order given by the Narayana numbers. We find natural ``creation and annihilation operators'' which allow us to define a QFT-type basis of each $SFH$ vector space, consisting of contact elements. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams. We prove that contact elements are in bijective correspondence with comparable pairs of basis elements with respect to a certain partial order, and in a natural and explicit way. The details of this description have intrinsic contact-topological meaning, and give rise to interesting category-theoretic, simplicial and algebraic structures.
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