Knots with infinitely many non-characterizing slopes

TL;DR

This paper demonstrates that certain knots, including $6_3$, have infinitely many non-characterizing slopes using annulus twist techniques, and explores applications and classifications related to annulus presentations.

Contribution

It introduces new techniques involving annulus twists to identify knots with infinitely many non-characterizing slopes and classifies knots with special annulus presentations up to 8 crossings.

Findings

$6_3$ has infinitely many non-characterizing slopes

Multiple knots up to 8 crossings have infinitely many non-characterizing slopes

Introduces the concept of trivial annulus twists and their applications

Abstract

Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$, $8_3$, $8_4$, $8_6$, $8_7$, $8_9$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$,$8_{20}$ and $8_{21}$ have infinitely many non-characterizing slopes. We also introduce the notion of trivial annulus twists and give some possible applications. Finally, we completely determine which knots have special annulus presentations up to 8-crossings.