Width is not additive

Abstract

We develop a construction suggested by Scharlemann and Thompson to obtain an infinite family of pairs of knots $K_{\alpha }$ and $K_{\alpha }^{\prime }$ so that $w(K_α # K'_α)=max{w(K_α), w(K'_α)}$. 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.

48 pages, 25 figures

References in corpus (1)

• Flipping bridge surfaces and bounds on the stable bridge number

Cited by in corpus (8)

• High Distance Bridge Surfaces
• Additive invariants for knots, links and graphs in 3-manifolds
• Width of a satellite knot and its companion
• Knots with compressible thin levels
• Trunk of Satellite and Companion Knots
• An Algorithmic Definition of Gabai Width
• Combinatorial minimal surfaces in pseudomanifolds
• Rectangle condition for compression body and 2-fold branched covering