Linear lambda terms as invariants of rooted trivalent maps

TL;DR

This paper establishes a conceptual correspondence between linear lambda calculus and rooted trivalent maps on surfaces, providing a new perspective that links lambda terms with topological graph invariants and combinatorial enumeration.

Contribution

It offers a simple, diagrammatic account of the lambda map correspondence, reconstructs maps from lambda terms via a Tutte-style recurrence, and reformulates the Four Color Theorem in lambda calculus terms.

Findings

Linear lambda terms serve as invariants for rooted trivalent maps.

Enumeration of bridgeless rooted trivalent maps using lambda calculus.

Reformulation of the Four Color Theorem through lambda calculus typing.

Abstract

The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between $\alpha$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the "easy" direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.