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

TL;DR

This paper explores the deep connections between knot theory, quantum mechanics, and logic by providing a new diagrammatic perspective on the Temperley-Lieb algebra, revealing its computational and logical significance.

Contribution

It offers a fully abstract, planar description of the Temperley-Lieb category using Geometry of Interaction, and introduces a planar lambda-calculus for diagrammatic computation.

Findings

Develops a planar version of Geometry of Interaction.

Provides a diagrammatic interpretation of computation as geometric simplification.

Shows how the Temperley-Lieb algebra underpins a planar lambda-calculus.

Abstract

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.