Asymmetric L-space knots

TL;DR

This paper constructs the first examples of asymmetric L-space knots in $S^3$, demonstrating hyperbolic knots with specific surgery properties and trivial isometry groups, expanding the understanding of knot symmetries.

Contribution

It introduces a novel construction of asymmetric L-space knots in $S^3$ and lens spaces, showing they can have trivial symmetry groups and specific surgery characteristics.

Findings

First examples of asymmetric L-space knots in $S^3$

Knots with surgeries related to strongly invertible links

Knots with trivial isometry groups

Abstract

We construct the first examples of asymmetric L-space knots in $S^3$. More specifically, we exhibit a construction of hyperbolic knots in $S^3$ with both (i) a surgery that may be realized as a surgery on a strongly invertible link such that the result of the surgery is the double branched cover of an alternating link and (ii) trivial isometry group. In particular, this produces L-space knots in $S^3$ which are not strongly invertible. The construction also immediately extends to produce asymmetric L-space knots in any lens space, including $S^1 \times S^2$.