On weak reducing disks and disk surgery

TL;DR

The paper constructs an example in 3-sphere showing that weak reducibility of disks is not preserved under disk surgery, challenging assumptions about disk properties in knot theory.

Contribution

It provides the first explicit example demonstrating that weak reducibility is not maintained after disk surgery in unknot positions.

Findings

Weak reducing disks can lose their property after surgery.

Disk surgery does not always preserve weak reducibility.

Counterexample in 8-bridge position of the unknot.

Abstract

Let $K$ be an unknot in $8$-bridge position in the $3$-sphere. We give an example of a pair of weak reducing disks $D_1$ and $D_2$ for $K$ such that both disks obtained from $D_i$ ($i = 1, 2$) by a surgery along any outermost disk in $D_{3-i}$, cut off by an outermost arc of $D_i \cap D_{3-i}$ in $D_{3-i}$, are not weak reducing disks, i.e. the property of weak reducibility of compressing disks is not preserved by a disk surgery.