On the additivity of knot width. (English) Zbl 1207.57016

Scharlemann, Martin; Thompson, Abigail

Gordon, Cameron (ed.) et al., Proceedings of the Casson Fest. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences, Fayetteville, AR, USA, April 10–12, 2003 and the conference on the topology of manifolds of dimensions 3 and 4, Austin, TX, USA, May 19–21, 2003. Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 135-144 (2004).

Summary: It has been conjectured that the geometric invariant of knots in 3-space called the width is nearly additive. That is, letting \(\omega (K) \in N\) denote the width of a knot \(K \in S^3\), the conjecture is that \(\omega (K \# K') = \omega (K) + \omega (K') - 2\). We give an example of a knot \(K_1\) so that for \(K_2\) any 2-bridge knot, it appears that \(\omega (K_1 \# K_2) = \omega (K_1)\), contradicting the conjecture.
For the entire collection see [Zbl 1066.57002].

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

Haken surfaces