We analyze symmetries, hidden symmetries and commensurability classes of
–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
–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.
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