Abstract
Menasco showed that a closed incompressible surface in the complement of a non-split prime alternating link in S3 contains a circle isotopic in the link complement to a meridian of the links. Based on this result, he was able to argue the hyperbolicity of non-split prime alternating links in S3. Adams et al. showed that if F ⊂ S × I L is an essential torus, then F contains a circle which is isotopic in S × I \ L to a meridian of L. The author generalizes his result as follows: Let S be a closed orientable surface, L be a fully alternating link in S × I. If F ⊂ S × I \ L is a closed essential surface, then F contains a circle which is isotopic in S × I \ L to a meridian of L.
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References
Adams, C., Toroidally alternating knots and links, Topology, 33(2), 1994, 353–369.
Adams, C., Generalized augmented alternating links and hyperbolic volumes, Algebr. Geom. Topol., 17(6), 2017, 3375–3397.
Adams, C., Albors-Riera, C., Haddock, B., et al., Hyperbolicity of links in thickened surfaces, Topology Appl., 256, 2019, 262–278.
Adams, C., Brock, J., Bugbee, J., et al., Almost alternating links, Topology Appl., 46(2), 1992, 151–165.
Bonahon, F. and Siebenmann, L., New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots, http://www-bcf.usc.edu/fbonahon/Research/Preprints/BonSieb.pdf
Fa, H., Incompressible pairwise incompressible surfaces in almost alternating knot complements, Topology Appl., 80(3), 1997, 239–249.
Hayashi, C., Link with alternating diagrams on closed surfaces of positive genus, Math. Proc. Camb. Phil. Soc., 117(1), 1995, 113–128.
Lozanoand, M. and Przytycki, J., Incompressible surfaces in the exterior of a closed 3-braid I, surfaces with horizontal boundary components, Math. Proc. Camb. Phil. Soc., 98, 1985, 275–299.
Menasco, W., Closed incompressible surfaces in alternating knot and link complements, Topology, 33(2), 1984, 353–369.
Oertel, U., Closed incompressible surfaces in complements of star links, Pacific J. Math., 111(1), 1984, 209–230.
Waldhausen, F., On irreducible 3-manifolds which are sufficiently large, Ann. Math., 87, 1968, 56–88.
Acknowledgement
I would like to thank William Menasco for his suggestions and many helpful discussions.
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Lin, W. The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements. Chin. Ann. Math. Ser. B 45, 73–80 (2024). https://doi.org/10.1007/s11401-024-0004-x
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DOI: https://doi.org/10.1007/s11401-024-0004-x