# A note on chirally cosmetic surgery on cable knots

## TL;DR

This paper investigates conditions under which cable and iterated torus knots do not admit chirally cosmetic surgeries, providing new restrictions based on knot structure and JSJ decomposition.

## Contribution

It establishes new non-existence results for chirally cosmetic surgeries on cable and iterated torus knots, extending previous understanding in knot theory.

## Key findings

- $(p,q)$-cable knots with $q
eq 2$ do not admit chirally cosmetic surgery.
- $(p,q)$-cable knots with $q=2$ do not admit chirally cosmetic surgery under certain JSJ conditions.
- Most iterated torus knots, except $(2,p)$-torus knots, do not admit chirally cosmetic surgery.

## Abstract

We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery as long as the JSJ piece of knot exterior does not contain $(2,r)$-torus exterior. We also show that an iterated torus knot other than $(2,p)$-torus knot does not admit chirally cosmetic surgery.