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In this paper we analyze symmetries, hidden symmetries, and commensurabilityclasses of $(\epsilon, d_L)$-twisted knot complements, which are thecomplements of knots that have a sufficiently large number of twists in each oftheir twist regions. These knot complements can be constructed via long Dehnfillings on fully augmented links complements. We show that such knotcomplements have no hidden symmetries, which implies that there are at most twoother knot complements in their respective commensurability classes. Under mildadditional hypotheses, we show that these knots have at most four(orientation-preserving) symmetries and are the only knot complements in theirrespective commensurability classes. Finally, we provide an infinite family ofexplicit examples of $(\epsilon, d_L)$-twisted knot complements that are theunique knot complements in their respective commensurability classes obtainedby filling a fully augmented link with four crossing circles.

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