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

TL;DR

This paper investigates the symmetries and hidden symmetries of a class of twisted knot complements, establishing their symmetry properties and uniqueness within their commensurability classes, with explicit examples provided.

Contribution

It demonstrates that these twisted knot complements lack hidden symmetries and are often unique in their commensurability classes, with explicit constructions and symmetry bounds.

Findings

No hidden symmetries in these knot complements

At most two other knots in the same commensurability class

Explicit examples of unique knot complements in their classes

Abstract

In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(\epsilon, d_L)$-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 links 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 $(\epsilon, d_L)$-twisted knot complements that are the unique knot complements in their respective commensurability classes obtained by filling a fully augmented link with four crossing circles.