Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics

Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics: Knot Theory, Categorical Quantum Mechanics, and Logic and Computation. We shall focus in particular on the following two topics: - The Temperley-Lieb algebra has always hitherto been presented as a quotient of some sort: either algebraically by generators and relations as in Jones' original presentation, or as a diagram algebra modulo planar isotopy as in Kauffman's presentation. We shall use tools from Geometry of Interaction, a dynamical interpretation of proofs under Cut Elimination developed as an off-shoot of Linear Logic, to give a direct description of the Temperley-Lieb category -- a "fully abstract presentation", in Computer Science terminology. This also brings something new to the Geometry of Interaction, since we are led to develop a planar version of it, and to verify that the interpretation of Cut-Elimination (the "Execution Formula", or "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification -- "yanking lines straight". We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category.