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

  • SEBASTIAN BAADER (a1) and ALEXANDRA KJUCHUKOVA (a2)
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.

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[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.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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