Graphic lambda calculus and knot diagrams, arXiv:1211.1604

Abstract

This version: 07.11.2012 In [5] was proposed a graphic lambda calculus formalism, which has sectors corre-sponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus. 1 Quick introduction and references The graphic lambda calculus [5] is a formalism based on local or global moves acting on locally planar trivalent graphs. In the mentioned paper we showed that ”sectors ” of this calculus are equivalent with untyped lambda calculus or with emergent algebras. (The formalism of emergent algebras [2] [3] evolved from differential calculus on metric spaces with dilations [1].) For all the relevant notions and results consult [5] for the graphic lambda calculus, [4] for λ-Scale calculus (a first proposal of a calculus containing both untyped lambda calculus and emergent algebras). Those interested in metric spaces with dilations and their applications may consult the course notes [6] on sub-riemannian geometry from intrinsic viewpoint. For a larger view on the half-dreamed subject of ”computing with space ” see [7], where a formalism for emergent algebras based on decorated knot diagrams for emergent algebras was proposed. There we called ”computing with space ” the various manipulations of these diagrams with geometric content. (The interested reader may browse various notes at [8], which are used as a repository and discussion place for the subject.