Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:15:44.231Z Has data issue: false hasContentIssue false
  • English
  • Français

EXCHANGE MOVES AND NONCONJUGATE BRAID REPRESENTATIVES OF KNOTS

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  20 May 2019

REIKO SHINJO  and
ALEXANDER STOIMENOW
REIKO SHINJO
Affiliation:
School of Science and Engineering, Kokushikan University, 4-28-1 Setagaya, Setagaya-ku, Tokyo 154-8515, Japan email reiko@kokushikan.ac.jp
ALEXANDER STOIMENOW
Affiliation:
School of General Studies, Gwangju Institute of Science and Technology, Gwangju 61005, Korea email stoimeno@stoimenov.nethttp://stoimenov.net/stoimeno/homepage/
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for $n\geqslant 4$, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move.

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 20F36: Braid groups; Artin groups 57M27: Invariants of knots and 3-manifolds
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 298 - 321
Check for updates
Copyright
© 2019 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, J. W., A lemma on systems of knotted curves, Proc. Natl. Acad. Sci. USA 9 (1923), 93–95.10.1073/pnas.9.3.93Google Scholar
Artin, E., Theory of braids, Ann. of Math. (2) 48 (1947), 101–126.10.2307/1969218Google Scholar
Birman, J. S., Non conjugate braids can define isotopic knots, Comm. Pure Appl. Math. 22 (1969), 239–242.10.1002/cpa.3160220207Google Scholar
Birman, J. S. and Menasco, W. W., Studying links via closed braids VI. A nonfiniteness theorem, Pacific J. Math. 156(2) (1992), 265–285.10.2140/pjm.1992.156.265Google Scholar
Birman, J. S. and Menasco, W. W., Studying links via closed braids III. Classifying links which are closed 3-braids, Pacific J. Math. 161(1) (1993), 25–113.10.2140/pjm.1993.161.25Google Scholar
Burde, G. and Zieschang, H., Knots, de Gruyter, Berlin, 1986.Google Scholar
Fukunaga, E., Infinite sequences of braids representing a single link type, “Topology of Knots V”, Waseda University, proceedings (2002).Google Scholar
Fukunaga, E., An infinite sequence of conjugacy classes in the $4$-braid group representing a torus link of type (2, k), Preprint.Google Scholar
Garside, F., The braid group and other groups, Q. J. Math. Oxford 20 (1969), 235–264.10.1093/qmath/20.1.235Google Scholar
Hoste, J., The first coefficient of the Conway polynomial, Proc. Amer. Math. Soc. 95(2) (1985), 299–302.10.1090/S0002-9939-1985-0801342-XGoogle Scholar
Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335–388.10.2307/1971403Google Scholar
Kawauchi, A., A survey of Knot Theory, Birkhäuser, Basel-Boston-Berlin, 1991.Google Scholar
Morton, H. R., Infinitely many fibred knots having the same Alexander polynomial, Topology 17(1) (1978), 101–104.10.1016/0040-9383(78)90016-2Google Scholar
Morton, H. R., Threading knot diagrams, Math. Proc. Camb. Philos. Soc. 99 (1986), 247–260.10.1017/S0305004100064161Google Scholar
Murakami, H. and Nakanishi, Y., On a certain move generating link-homology, Math. Ann. 284 (1989), 75–89.10.1007/BF01443506Google Scholar
Murasugi, K. and Thomas, R. S. D., Isotopic closed nonconjugate braids, Proc. Amer. Math. Soc. 33 (1972), 137–139.10.1090/S0002-9939-1972-0292061-0Google Scholar
Rolfsen, D., Knots and Lnks, Publish or Perish, 1976.Google Scholar
Shinjo, R., “An infinite sequence of non-conjugate 4-braids representing the same knot of braid index 4”, Intelligence of Low Dimensional Topology 2006, Ser. Knots Everything, 40, 293–297. World Sci. Publ., 2007.10.1142/9789812770967_0037Google Scholar
Shinjo, R., Non-conjugate braids whose closures result in the same knot, J. Knot Theory Ramifications 19(1) (2010), 117–124.10.1142/S0218216510007735Google Scholar
Shinjo, R. and Stoimenow, A., Infinitely many non-conjugate braid representatives of links, to appear.Google Scholar
Stoimenow, A., On non-conjugate braids with the same closure link, J. Geometry 96(1) (2010), 179–186.10.1007/s00022-010-0029-zGoogle Scholar
Stoimenow, A., Non-conjugate braids with the same closure link from density of representations, J. Math. Pures Appl. 94(5) (2010), 470–496.10.1016/j.matpur.2010.08.003Google Scholar
Stoimenow, A., Tabulating and distinguishing mutants, Internat. J. Algebra Comput. 20(4) (2010), 525–559.10.1142/S0218196710005789Google Scholar
Stoimenow, A., Knot data tables, http://stoimenov.net/stoimeno/homepage/ptab/braid.out.Google Scholar