Width is not additive

Blair, Ryan¹ ; Tomova, Maggy²

¹ Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 709 South 33rd Street, Philadelphia, PA 19104-6395, United States
² Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, United States

Geometry & topology, Tome 17 (2013) no. 1, pp. 93-156.

Résumé

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 $w\left(K\#K^{\prime }\right)<w\left(K\right)+w\left(K^{\prime }\right)-2$ 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.

Détail

DOI : 10.2140/gt.2013.17.93

Classification : 57M25, 57M27, 57M50
Keywords: width, thin position, connected sum, high distance surface

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