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.
From: Christopher-Lloyd Simon [view email]
[v1]
Fri, 13 Oct 2023 01:30:10 UTC (485 KB)
[v2]
Fri, 3 Nov 2023 02:26:05 UTC (500 KB)
[v3]
Sat, 20 Jan 2024 23:01:26 UTC (185 KB)
[v4]
Sun, 13 Sep 2026 12:04:41 UTC (180 KB)
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