Properties of knots preserved by cabling

TL;DR

This paper investigates how certain geometric properties of knots, such as width and smallness, are preserved under cabling operations, establishing conditions under which thin positions and minimal bridge positions are maintained.

Contribution

It provides new insights into the invariance of knot properties like width and smallness under cabling, linking thin positions of knots and their cables.

Findings

Thin position of a cable knot corresponds to an 'obvious' cabling of the thin position of the original knot.

A knot is meridionally small if and only if its cable is meridionally small.

If the original knot's non-minimal bridge positions are stabilized, the same holds for its cable.

Abstract

We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is stabilized, then the same is true for K.