Seifert-fibre-edge conjecture for one-vertex triangulations of small Seifert fibre spaces
A Seifert fibre space is a 3-manifold fibred by circles in the Seifert sense, and a small Seifert fibre space is one that contains no embedded two-sided incompressible surfaces. An edge is isotopic to a Seifert fibre when the embedded closed curve it realizes is isotopic to one of the circle fibres.
Seifert-fibre-edge conjecture. Every one-vertex triangulation of a small Seifert fibre space has an edge isotopic to a Seifert fibre.
Small Seifert fibre spaces are important pieces in JSJ and geometric decompositions, but their lack of embedded two-sided incompressible surfaces makes them difficult to study. The paper presents this as an unresolved conjecture motivating its search for counterexamples.
References
Primary source
Benjamin A. Burton and Alexander He, “Finding large counterexamples by selectively exploring the Pachner graph”, arXiv:2303.06321 (2024).
Additional references
3 papers in this index state this conjecture (2007–2023). The statement above is taken from the most recent of them; the others are arXiv:1202.4142, arXiv:math/0703276.
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