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

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