Volume 17, issue 1 (2013)

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Width is not additive

Ryan Blair and Maggy Tomova

Geometry & Topology 17 (2013) 93–156
Abstract

We develop the construction suggested by Scharlemann and Thompson in [Proc. of the Casson Fest. (2004) 135-144] to obtain an infinite family of pairs of knots ${K}_{\alpha }$ and ${K}_{\alpha }^{\prime }$ so that $w\left({K}_{\alpha }#{K}_{\alpha }^{\prime }\right)=max\left\{w\left({K}_{\alpha }\right),w\left({K}_{\alpha }^{\prime }\right)\right\}$. This is the first known example of a pair of knots such that $w\left(K#{K}^{\prime }\right) and it establishes that the lower bound $w\left(K#{K}^{\prime }\right)\ge max\left\{w\left(K\right),w\left({K}^{\prime }\right)\right\}$ obtained in Scharlemann and Schultens [Trans. Amer. Math. Soc. 358 (2006) 3781-3805] 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.

Keywords
width, thin position, connected sum, high distance surface
Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57M50
Publication
Received: 18 June 2010
Revised: 25 March 2012
Accepted: 16 July 2012
Published: 25 February 2013
Proposed: Cameron Gordon
Seconded: David Gabai, Ronald Stern
Authors
 Ryan Blair Department of Mathematics University of Pennsylvania David Rittenhouse Lab 709 South 33rd Street Philadelphia, PA 19104-6395 United States http://www.math.upenn.edu/~ryblair Maggy Tomova Department of Mathematics University of Iowa 14 MacLean Hall Iowa City, IA 52242-1419 United States http://www.math.uiowa.edu/~mtomova