Hyperbolic knots given by positive braids with at least two full twists

de Paiva, Thiago

doi:10.1090/proc/16035

Hyperbolic knots given by positive braids with at least two full twists
HTML articles powered by AMS MathViewer

by Thiago de Paiva PDF

Proc. Amer. Math. Soc. 150 (2022), 5449-5458 Request permission

Abstract:

We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots.

References

Joan Birman and Ilya Kofman, A new twist on Lorenz links, J. Topol. 2 (2009), no. 2, 227–248. MR 2529294, DOI 10.1112/jtopol/jtp007
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
Joan S. Birman and William W. Menasco, Studying links via closed braids. I. A finiteness theorem, Pacific J. Math. 154 (1992), no. 1, 17–36. MR 1154731, DOI 10.2140/pjm.1992.154.17
Joan S. Birman and William W. Menasco, Special positions for essential tori in link complements, Topology 33 (1994), no. 3, 525–556. MR 1286930, DOI 10.1016/0040-9383(94)90027-2
Thiago de Paiva, Unexpected essential surfaces among exteriors of twisted torus knots, arXiv:2110.09873, to appear in Algebr. Geom. Topol., 2021.
Thiago de Paiva and Jessica S. Purcell, Satellites and Lorenz knots, arXiv:2103.09500, 2021.
John Charles Dean, Hyperbolic knots with small Seifert-fibered Dehn surgeries, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–The University of Texas at Austin. MR 2694392
Patrick Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), no. 1, 115–150. MR 1214782, DOI 10.1090/S0002-9947-1994-1214782-4
John Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), no. 1, 97–108. MR 896009, DOI 10.1090/S0002-9947-1987-0896009-2
Tetsuya Ito, Braid ordering and the geometry of closed braid, Geom. Topol. 15 (2011), no. 1, 473–498. MR 2788641, DOI 10.2140/gt.2011.15.473
Sangyop Lee, Torus knots obtained by twisting torus knots, Algebr. Geom. Topol. 15 (2015), no. 5, 2819–2838. MR 3426694, DOI 10.2140/agt.2015.15.2819
Sangyop Lee, Knot types of twisted torus knots, J. Knot Theory Ramifications 26 (2017), no. 12, 1750074, 7. MR 3718275, DOI 10.1142/S0218216517500742
Sangyop Lee, Satellite knots obtained by twisting torus knots: hyperbolicity of twisted torus knots, Int. Math. Res. Not. IMRN 3 (2018), 785–815. MR 3801447, DOI 10.1093/imrn/rnw255
Sangyop Lee, Composite knots obtained by twisting torus knots, Int. Math. Res. Not. IMRN 18 (2019), 5744–5776. MR 4012126, DOI 10.1093/imrn/rnx282
Sangyop Lee and Thiago de Paiva, Torus knots obtained by negatively twisting torus knots, arXiv:2108.10641, 2021.
Jérôme E. Los, Knots, braid index and dynamical type, Topology 33 (1994), no. 2, 257–270. MR 1273785, DOI 10.1016/0040-9383(94)90014-0
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
R. F. Williams, The braid index of generalized cables, Pacific J. Math. 155 (1992), no. 2, 369–375. MR 1178031, DOI 10.2140/pjm.1992.155.369