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Hyperbolicity of links in thickened surfaces. (English) Zbl 1412.57002

The main result of this paper is that if \(S\) is a closed orientable surface, then a prime, fully alternating link \(L\) in the thickened surface \(S\times I\) is hyperbolic, that is, the complement \(S\times I \backslash L\) supports a hyperbolic metric. By definition \(L\) is prime in \(S\times I\) if there does not exist an essential twice-punctured sphere in \(S\times I \backslash L\) such that both punctures are created by \(L\), and it is fully alternating if its projection onto \(S\) is alternating on \(S\) and the interior of the closure of every complementary region is an open disk.
The proof of the main result builds on techniques in [W. Menasco, Topology 23, 37–44 (1984; Zbl 0525.57003)]. As in Menasco’s paper, the authors also provide an easy way to check whether a link is prime.
In addition they extend their main result to prime fully alternating links \(L\) in a neighborhood of an essential closed surface \(S\) in an orientable, hyperbolic 3-manifold \(M\). Some related recent papers are [A. Champanerkar et al., J. Lond. Math. Soc., II. Ser. 99, No. 3, 807–830 (2019; Zbl 1456.57004)] and [J. A. Howie and J. S. Purcell, Trans. Am. Math. Soc. 373, No. 4, 2349–2397 (2020; Zbl 1441.57007)].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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References:

[1] Adams, C., Toroidally alternating knots and links, Topology, 33, 2, 353-369 (1994) · Zbl 0839.57004
[2] Adams, C.; Calderon, A.; Meyer, N., Generalized bipyramids and hyperbolic volumes of alternating \(k\)-uniform tiling links (2016), arXiv preprint
[3] Adams, C.; Fleming, T.; Levin, M.; Turner, A. M., Crossing number of alternating knots in \(S \times I\), Pac. J. Math., 203, 1, 1-22 (2002) · Zbl 1051.57006
[4] Agol, I., The Virtual Haken Conjecture, Doc. Math., 18, 1045-1087 (2013), (with appendix by Agol, Groves and Manning) · Zbl 1286.57019
[5] Champanerkar, A.; Kofman, I.; Purcell, J., Geometry of biperiodic alternating links (2018), to appear in J. London Math. Soc., arXiv preprint · Zbl 1456.57004
[6] Hatcher, A., Notes on Basic 3-Manifold Topology, 1-72 (2000)
[7] Hayashi, C., Links with alternating diagrams on closed surfaces of positive genus, Math. Proc. Camb. Philos. Soc., 117, 113-128 (1995) · Zbl 0844.57005
[8] Howie, J.; Purcell, J., Geometry of alternating links on surfaces (2017), arXiv preprint
[9] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 395-407 (1987) · Zbl 0622.57004
[10] Menasco, W., Closed incompressible surfaces in alternating knot and link complements, Topology, 23, 1, 37-44 (1984) · Zbl 0525.57003
[11] Murasugi, K., The Jones polynomial and classical conjectures in Knot theory, Topology, 26, 187-194 (1987) · Zbl 0628.57004
[12] Thistlethwaite, M. B., A spanning tree expansion of the Jones polynomial, Topology, 26, 297-309 (1987) · Zbl 0622.57003
[13] W. Thurston, The Geometry and Topology of 3-manifolds, Princeton Notes.; W. Thurston, The Geometry and Topology of 3-manifolds, Princeton Notes. · Zbl 0483.57007
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