Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913201682120704 |
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| author | Simon, Christopher-Lloyd |
| author_facet | Simon, Christopher-Lloyd |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_08806 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars Simon, Christopher-Lloyd Geometric Topology Combinatorics 57M15, 05C10, 57K12, 81Q30 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. |
| title | Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars |
| topic | Geometric Topology Combinatorics 57M15, 05C10, 57K12, 81Q30 |
| url | https://arxiv.org/abs/2310.08806 |