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

Abstract

We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the $\left(1+1\right)$–dimensional topological quantum field theory defined by Honda, Kazez and Matić in [arXiv:0807.2431]. The ${\mathbb{Z}}_2$ SFH of these solid tori forms a “categorification of Pascal’s triangle”, and contact structures correspond bijectively to chord diagrams, or sets of disjoint properly embedded arcs in the disc. Their contact elements are distinct and 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 algebraic and combinatorial structures in this description have intrinsic contact-topological meaning.

In particular, the QFT–basis of SFH 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.