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Symmetries and hidden symmetries of $(\epsilon, d_L)$–twisted knot complements

Neil R Hoffman, Christian Millichap and William Worden

Algebraic & Geometric Topology 22 (2022) 601–656

We analyze symmetries, hidden symmetries and commensurability classes of (𝜖,dN)–twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmented link complements. We show that such knot complements have no hidden symmetries, which implies that there are at most two other knot complements in their respective commensurability classes. Under mild additional hypotheses, we show that these knots have at most four (orientation-preserving) symmetries and are the only knot complements in their respective commensurability classes. Finally, we provide an infinite family of explicit examples of (𝜖,dN)–twisted knot complements that are the unique knot complements in their respective commensurability classes obtained by filling a fully augmented link complement with four crossing circles.

commensurability classes of knot complements, hidden symmetries, orbifolds, Dehn filling, fully augmented links
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27, 57M50
Received: 23 September 2019
Revised: 29 November 2020
Accepted: 30 December 2020
Published: 3 August 2022
Neil R Hoffman
Department of Mathematics
Oklahoma State University
Stillwater, OK
United States
Christian Millichap
Department of Mathematics
Furman University
Greenville, SC
United States
William Worden
Department of Mathematics
Rice University
Houston, TX
United States