Generalized crossing changes in satellite knots

TL;DR

This paper investigates conditions under which satellite knots admit generalized cosmetic crossing changes, establishing links to pattern knots and hyperbolic knots, and constraining possible knot types for such changes.

Contribution

It proves that satellite knots with certain crossing changes relate to pattern knots with the same property and shows hyperbolic knots are necessary for large-order crossing changes.

Findings

Satellite knots with crossing change of order |q| ≥ 6 relate to pattern knots with the same crossing change.

Prime satellite knots with fibered pattern knots cannot admit large-order crossing changes.

Any knot with a large-order crossing change must be hyperbolic.

Abstract

We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered cannot admit a generalized cosmetic crossing changes of order q with |q| \geq 6. We also show that if there is any knot admitting a generalized cosmetic crossing change of order q with |q| \geq 6, then there must be such a knot which is hyperbolic.