
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 and 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 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
