Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:49:11.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetric quotients of knot groups and a filtration of the Gordian graph

Part of: Low-dimensional topology

Published online by Cambridge University Press:  10 April 2019

SEBASTIAN BAADER  and
ALEXANDRA KJUCHUKOVA
SEBASTIAN BAADER
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. e-mail: sebastian.baader@math.unibe.ch
ALEXANDRA KJUCHUKOVA
Affiliation:
Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Dr, Madison, WI 53703, U.S.A. e-mail: kjuchukova@math.wise.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

MSC classification

Primary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 141 - 148
Copyright
© Cambridge Philosophical Society 2019

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

REFERENCES

Baader, S.. Note on crossing changes. Quart. J. Math. 57 (2006), 139–142.CrossRefGoogle Scholar
Cahn, P. and Kjuchukova, A.. Linking numbers in three-manifolds. arXiv: 1611.10330.Google Scholar
Crowell, R. H. and Fox, R. H.. Introduction to Knot Theory, reprint of the 1963 original (Springer-Verlag, 1977).CrossRefGoogle Scholar
Fox, R. H.. Metacyclic invariants of knots and links. Canad. J. of Math. 22 (1970), 193–201.CrossRefGoogle Scholar
Gambaudo, J. M. and Ghys, E.. Braids and signatures. Bull. Soc. Math. France 133 (2005), no. 4, 541–579.CrossRefGoogle Scholar
Hirasawa, M. and Uchida, Y.. The Gordian complex of knots. J. Knot Theory Ramifications 11 (2002), no. 3, 36–368.CrossRefGoogle Scholar
Kauffman, L. H.. Knots and physics, 3rd edition Series on Knots and Everything 1 (World Scientific Publishing Co., River Edge, NJ 2001).CrossRefGoogle Scholar
Kirby, R. (editor). Problems in low-dimensional topology in Proceedings of Georgia Topology Conference, Part 2 (Citeseer, 1995).Google Scholar
Murakami, H.. Some metrics on classical knots. Math. Ann. 270 (1985), 35–45.CrossRefGoogle Scholar
Perko, M.. An invariant of certain knots, Undergraduate Thesis (1964).Google Scholar
Perko, M.. On the classification of knots, Proc. Amer. Math. Soc 45 (1974), 262–266.CrossRefGoogle Scholar
Perko, M.. Visualising linking numbers in 3-coloured knot covers. Private correspondence (2018).Google Scholar
Reidemeister, K.. Knoten und Verkettungen. Mathematische Zeitschrift 29 (1929), no. 1, 713–729.CrossRefGoogle Scholar
Scharlemann, M.. Crossing changes, Knot theory and its applications. Chaos Solitons Fractals 9 (1998), no. 4–5, 693–704.CrossRefGoogle Scholar
Schubert, H.. Über eine numerische Knoteninvariante. Math. Z. 61 (1954), 245–288.CrossRefGoogle Scholar