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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
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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

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