Cyclic branched covers of alternating knots
Annales Henri Lebesgue, Volume 4 (2021), pp. 811-830.

Metadata

Keywords Alternating knots, prime knots, cyclic branched covers of knots, periodic symmetries of knots

Abstract

For any integer n>2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.


References

[Ban30] Bankwitz, Carl Über die Torsionszahlen der alternierenden Knoten, Math. Ann., Volume 103 (1930), pp. 145-161 | DOI | MR | Zbl

[BLP05] Boileau, Michel; Leeb, Bernhard; Porti, Joan Geometrization of 3-dimensional orbifolds, Ann. Math., Volume 162 (2005) no. 1, pp. 195-290 | DOI | MR | Zbl

[BMP03] Boileau, Michel; Maillot, Sylvain; Porti, Joan Three-dimensional orbifolds and their geometric structures, Panoramas et Synthèses, 15, Société Mathématique de France, 2003 | MR | Zbl

[Boy19] Boyle, Keegan Odd order group actions on alternating knots (2019) (https://arxiv.org/abs/1906.04308)

[BP01] Boileau, Michel; Porti, Joan Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti, Astérisque, 272, Société Mathématique de France, 2001 | Numdam | Zbl

[BP08] Boileau, Michel; Paoluzzi, Luisa On cyclic branched coverings of prime knots, J. Topol., Volume 1 (2008) no. 3, pp. 557-583 | DOI | MR | Zbl

[BZ03] Burde, Gerhard; Zieschang, Heiner Knots, De Gruyter Studies in Mathematics, 5, Walter de Gruyter, 2003 | Zbl

[CHK00] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, 2000 | MR | Zbl

[Con67] Conway, John H. An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon Press (1967), pp. 329-358 | Zbl

[CQH19] Costa, Antonio F.; Quach Hongler, Can V. Periodic projections of alternating knots (2019) (https://arxiv.org/abs/1905.13718)

[Cro04] Cromwell, Peter R. Knots and links, Cambridge University Press, 2004 | MR | Zbl

[Dun88] Dunbar, William D. Geometric orbifolds, Rev. Mat. Univ. Complutense Madr., Volume 1 (1988) no. 1-3, pp. 67-99 | MR | Zbl

[Gre13] Greene, Joshua E. Lattices, graphs, and Conway mutation, Invent. Math., Volume 192 (2013) no. 3, pp. 717-750 | DOI | MR | Zbl

[Gre17] Greene, Joshua E. Alternating links and definite surfaces, Duke Math. J., Volume 166 (2017) no. 11, pp. 2133-2151 | MR | Zbl

[How17] Howie, Joshua A. A characterisation of alternating knot exteriors, Geom. Topol., Volume 21 (2017) no. 4, pp. 2353-2371 | DOI | MR | Zbl

[HR85] Hodgson, Craig D.; Rubinstein, J. Hyam Involutions and isotopies of lens spaces, Knot theory and manifolds (Vancouver, 1983) (Rolfsen, Dale, ed.) (Lecture Notes in Mathematics), Volume 1144, Springer, 1985, pp. 60-96 | DOI | MR | Zbl

[HTW98] Hoste, Jim; Thistlethwaite, Morwen B.; Weeks, Jeff The first 1,701,936 knots, Math. Intell., Volume 20 (1998) no. 4, pp. 33-48 | DOI | MR | Zbl

[Jac80] Jaco, William Lectures on three-manifold topology, Regional Conference Series in Mathematics, 43, American Mathematical Society, 1980 | MR | Zbl

[KL99] Kirk, Paul; Livingston, Charles Twisted knot polynomials: inversion, mutation and concordance, Topology, Volume 38 (1999) no. 3, pp. 663-671 | DOI | MR | Zbl

[Koj86] Kojima, Sadayoshi Determining knots by branched covers, Low Dimensional Topology and Kleinian groups (London Mathematical Society Lecture Note Series), Volume 112, Cambridge University Press, 1986, pp. 193-207 | MR | Zbl

[KT57] Kinoshita, Shin’ichi; Terasaka, Hidetaka On unions of knots, Osaka J. Math., Volume 9 (1957), pp. 131-153 | MR | Zbl

[MA73] Montesinos-Amilibia, José M. Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Mat. Mex., Volume 18 (1973), pp. 1-32 | Zbl

[MB84] Morgan, John; Bass, Hyman The Smith conjecture, Pure and Applied Mathematics, 112, Academic Press Inc., 1984 | MR | Zbl

[Men84] Menasco, William W. Closed incompressible surfaces in alternating knot and link complements, Topology, Volume 23 (1984), pp. 37-44 | DOI | MR | Zbl

[Mil75] Milnor, John W. On the 3-dimensional Brieskorn manifolds M(p,q,r), Knots, groups and 3-manifolds (Papers dedicated to the memory of R. H. Fox) (Annals of Mathematics Studies), Volume 84, Princeton University Press, 1975, pp. 175-225 | DOI | Zbl

[MT93] Menasco, William W.; Thistlethwaite, Morwen B. The classification of alternating links, Ann. Math., Volume 138 (1993) no. 1, pp. 113-171 | DOI | MR | Zbl

[Mur58] Murasugi, Kunio On the Alexander polynomial of the alternating knot, Osaka J. Math., Volume 10 (1958), pp. 181-189 | MR | Zbl

[Nak81] Nakanishi, Yasutaka Primeness of links, Math. Semin. Notes, Kobe Univ., Volume 9 (1981), pp. 415-440 | MR | Zbl

[Neu70] Neumann, Walter D. S 1 -actions and the α-invariant of their involutions, Ph. D. Thesis, Bonn University, Germany (1970) (http://www.math.columbia.edu/~neumann/preprints/neumann011.pdf)

[NRL03] Núñez, Víctor; Ramírez-Losada, Enrique The trefoil knot is as universal as it can be, Topology Appl., Volume 130 (2003) no. 1, pp. 1-17 | DOI | MR | Zbl

[Pao05a] Paoluzzi, Luisa Hyperbolic knots and cyclic branched covers, Publ. Mat., Barc., Volume 49 (2005) no. 2, pp. 257-284 | DOI | MR | Zbl

[Pao05b] Paoluzzi, Luisa Three cyclic branched covers suffice to determine hyperbolic knots, J. Knot Theory Ramifications, Volume 14 (2005) no. 5, pp. 641-655 | DOI | MR | Zbl

[Rol76] Rolfsen, Dale Knots and links, Mathematical Lecture Series, 7, Publish or Perish Inc., 1976 | MR | Zbl

[Sak81] Sakuma, Makoto Periods of composite links, Math. Semin. Notes, Kobe Univ., Volume 9 (1981), pp. 445-452 | MR | Zbl

[Vir72] Viro, Oleg Y. Links, two-sheeted branching coverings and braids, Mat. Sb., N. Ser., Volume 87 (129) (1972), p. 216--228 | Zbl

[Vir76] Viro, Oleg Y. Non-projecting isotopies and knots with homeomorphic coverings, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Volume 66 (1976), pp. 133-147 (English transl. in J. Soviet Math.) | Zbl

[Zim98] Zimmermann, Bruno On hyperbolic knots with homeomorphic cyclic branched coverings, Math. Ann., Volume 311 (1998) no. 4, pp. 665-673 | DOI | MR | Zbl