Skip to main content Accessibility help

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



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.



Hide All
[1]Baader, S.. Note on crossing changes. Quart. J. Math. 57 (2006), 139142.
[2]Cahn, P. and Kjuchukova, A.. Linking numbers in three-manifolds. arXiv: 1611.10330.
[3]Crowell, R. H. and Fox, R. H.. Introduction to Knot Theory, reprint of the 1963 original (Springer-Verlag, 1977).
[4]Fox, R. H.. Metacyclic invariants of knots and links. Canad. J. of Math. 22 (1970), 193201.
[5]Gambaudo, J. M. and Ghys, E.. Braids and signatures. Bull. Soc. Math. France 133 (2005), no. 4, 541579.
[6]Hirasawa, M. and Uchida, Y.. The Gordian complex of knots. J. Knot Theory Ramifications 11 (2002), no. 3, 36368.
[7]Kauffman, L. H.. Knots and physics, 3rd edition Series on Knots and Everything 1 (World Scientific Publishing Co., River Edge, NJ 2001).
[8]Kirby, R. (editor). Problems in low-dimensional topology in Proceedings of Georgia Topology Conference, Part 2 (Citeseer, 1995).
[9]Murakami, H.. Some metrics on classical knots. Math. Ann. 270 (1985), 3545.
[10]Perko, M.. An invariant of certain knots, Undergraduate Thesis (1964).
[11]Perko, M.. On the classification of knots, Proc. Amer. Math. Soc 45 (1974), 262266.
[12]Perko, M.. Visualising linking numbers in 3-coloured knot covers. Private correspondence (2018).
[13]Reidemeister, K.. Knoten und Verkettungen. Mathematische Zeitschrift 29 (1929), no. 1, 713729.
[14]Scharlemann, M.. Crossing changes, Knot theory and its applications. Chaos Solitons Fractals 9 (1998), no. 4–5, 693704.
[15]Schubert, H.. Über eine numerische Knoteninvariante. Math. Z. 61 (1954), 245288.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed