Chord diagrams, contact-topological quantum field theory, and contact categories

In this thesis we study some interesting mathematics arising at the intersection of the studies of contact topology and sutured Floer homology. Although this work was originally motivated by the study of contact elements in sutured Floer homology, we also obtain results in pure contact topology.;We consider contact elements in the sutured Floer homology of solid tori, as part of the (1+1)-dimensional topological quantum field theory defined by Honda-Kazez-Matic. We find that the Z2 sutured Floer homology of solid tori with longitudinal sutures forms a "categorification of Pascal's triangle", a triangle of vector spaces. Contact structures on solid tori with longitudinal sutures correspond bijectively to chord diagrams, which are sets of disjoint properly embedded arcs in the disc; these may in turn be identified with contact elements. The contact elements form distinguished subsets of the vector spaces in the categorified Pascal's triangle, 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. We show that 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. We also prove numerous results about the structure of contact elements and investigate various algebraic structures which arise.;Our main theorem, describing how contact elements lie in sutured Floer homology, has a purely combinatorial interpretation, as a statement about chords on discs subject to a certain surgery and a single addition relation. The algebraic and combinatorial structures which naturally arise in this description have intrinsic contact-topological meaning.;In particular, the QFT-type basis of sutured Floer homology, and its partial order, have a natural interpretation in pure contact topology, related to the contact category of a disc: the partial order enables us to tell when the sutured solid cylinder obtained by "stacking" two chord diagrams has a tight contact structure. This leads us to extend Honda's notion of contact category to a "bounded" contact category, containing chord diagrams and contact structures which occur within a given contact solid cylinder. We compute this bounded contact category in certain cases. Moreover, the decomposition of a contact element into basis elements naturally gives a triple of contact structures on solid cylinders which we regard as a type of "distinguished triangle" in the contact category. We also use the algebraic structures arising among contact elements to extend the notion of contact category to a 2-category.