Width is not additive

TL;DR

This paper constructs specific knot pairs demonstrating that width is not additive under connected sum, providing counterexamples to previous bounds and exploring the relationship between thin and bridge positions.

Contribution

It introduces an infinite family of knot pairs where width behaves non-additively, challenging prior assumptions and bounds in knot theory.

Findings

First known example where width of connected sum is less than sum of widths minus two

Shows the lower bound for width of connected sum is optimal

Provides knots with more critical points in thin position than in bridge position

Abstract

We develop a construction suggested by Scharlemann and Thompson to obtain an infinite family of pairs of knots $K_{\alpha}$ and $K'_{\alpha}$ so that $w(K_{\alpha} # K'_{\alpha})=max{w(K_{\alpha}), w(K'_{\alpha})}$. This is the first known example of a pair of knots such that $w(K#K')<w(K)+w(K')-2$ and it establishes that the lower bound $w(K#K')\geq max{w(K),w(K')}$ obtained by Scharlemann and Schultens is best possible. Furthermore, the knots $K_{\alpha}$ provide an example of knots where the number of critical points for the knot in thin position is greater than the number of critical points for the knot in bridge position.