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

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

2009-10-14

arxiv.org/abs/0910.2737v1

In Mathematics of Quantum Computing and Technology, ed. G. Chen, L. Kauffman and S. Lomonaco. Taylor and Francis, 415--458, 20

Physics

Quantum Physics

45 pages

Scientific paper

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.

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