Geometry & Topology

Width is not additive

Ryan Blair and Maggy Tomova

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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 Kα and Kα so that w(Kα#Kα)= max{w(Kα),w(Kα)}. 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) max{w(K),w(K)} obtained in Scharlemann and Schultens [Trans. Amer. Math. Soc. 358 (2006) 3781-3805] is best possible. Furthermore, the knots Kα 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.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 93-156.

Received: 18 June 2010
Revised: 25 March 2012
Accepted: 16 July 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57M50: Geometric structures on low-dimensional manifolds

width thin position connected sum high distance surface


Blair, Ryan; Tomova, Maggy. Width is not additive. Geom. Topol. 17 (2013), no. 1, 93--156. doi:10.2140/gt.2013.17.93.

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