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We consider contact elements in the sutured Floer homology of solid tori withlongitudinal sutures, as part of the (1+1)-dimensional topological quantumfield theory defined by Honda--Kazez--Mati\'{c} in \cite{HKM08}. The $\Z_2$$SFH$ of these solid tori forms a "categorification of Pascal's triangle", andcontact structures correspond bijectively to chord diagrams, or sets ofdisjoint properly embedded arcs in the disc. Their contact elements aredistinct and form distinguished subsets of $SFH$ of order given by the Narayananumbers. We find natural "creation and annihilation operators" which allow usto define a QFT-type basis of $SFH$, consisting of contact elements. SuturedFloer homology in this case reduces to the combinatorics of chord diagrams. Weprove that contact elements are in bijective correspondence with comparablepairs of basis elements with respect to a certain partial order. The algebraicand combinatorial structures in this description have intrinsiccontact-topological meaning. In particular, the QFT-basis of $SFH$ and itspartial order have a natural interpretation in pure contact topology, relatedto the contact category of a disc: the partial order enables us to tell whenthe sutured solid cylinder obtained by "stacking" two chord diagrams has atight contact structure. This leads us to extend Honda's notion of contactcategory to a "bounded" contact category, containing chord diagrams and contactstructures which occur within a given contact solid cylinder. We compute thisbounded contact category in certain cases. Moreover, the decomposition of acontact element into basis elements naturally gives a triple of contactstructures on solid cylinders which we regard as a type of "distinguishedtriangle" in the contact category. We also use the algebraic structures arisingamong contact elements to extend the notion of contact category to a2-category.

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