Volume 4, issue 2 (2004)

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Mp-small summands increase knot width

Jacob Hendricks

Algebraic & Geometric Topology 4 (2004) 1041–1044
 arXiv: math.GT/0406072
Abstract

Scharlemann and Schultens have shown that for any pair of knots ${K}_{1}$ and ${K}_{2}$, $w\left({K}_{1}#{K}_{2}\right)\ge max\left\{w\left({K}_{1}\right),w\left({K}_{2}\right)\right\}$. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann–Schultens, Rieck–Sedgwick and Thompson, it is shown that for $K={#}_{i=1}^{n}{K}_{i}$ a connected sum of mp-small knots and ${K}^{\prime }$ any non-trivial knot, $w\left(K#{K}^{\prime }\right)>w\left(K\right)$.

Keywords
thin position, knot width
Primary: 57M25
Secondary: 57M27