Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars

TL;DR

This paper introduces a method to compute the minimal genus of filoops from interlace graphs, describes a canonical factorisation into spheric and toric sums, and develops grammars generating specific graph classes.

Contribution

It provides a novel computation of mg(G), a canonical factorisation of filoops, and unambiguous grammars for graphs with mg(G)=0, advancing understanding of filoops and their graph representations.

Findings

mg(G) determines minimal genus of filoops

Canonical factorisation into spheric and toric sums is established

Unambiguous grammars generate graphs with mg(G)=0

Abstract

A filoop is a generic immersion of a circle in a closed oriented surface, whose complement is a disjoint union of discs, considered up to orientation preserving diffeomorphisms. It gives rise to a chord diagram C which has an interlace graph G, called a chordiagraph. For a graph G with even degrees, we compute a quantity mg(G) which yields, for every chord diagram $C$ with interlace graph G, the minimal genus of filoops with chord diagram C. If mg(G)=0 then C admits exactly two framings of genus 0, corresponding to spheriloops. After recalling the Cunningham factorisation of connected graphs, we describe a canonical factorisation of filoops into spheric sums followed by toric sums, for which the genus is additive. This is analogous to the factorisation of compact connected 3-manifolds along spheres and tori. We describe unambiguous context-sensitive grammars generating the set of all graphs and with mg(G)=0 and deduce stability properties with respect to spheric and toric factorisations. Similar results hold for chordiagraphs with mg(G) = 0 and their corresponding spheriloops.