We consider contact elements in the sutured Floer homology
of solid tori with longitudinal sutures, as part of the
–dimensional
topological quantum field theory defined by Honda, Kazez and Matić in [arXiv:0807.2431].
The
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
–category.
