Companions of the unknot and width additivity

Ryan Blair; Maggy Tomova

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arXiv · 0908.4103

Companions of the unknot and width additivity

Ryan Blair·Maggy Tomova

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Abstract

It has been conjectured that for knots $K$ and $K'$ in $S^3$, $w(K#K')= w(K)+w(K')-2$. Scharlemann and Thompson have proposed potential counterexamples to this conjecture. For every $n$, they proposed a family of knots ${K^n_i}$ for which they conjectured that $w(B^n#K^n_i)=w(K^n_i)$ where $B^n$ is a bridge number $n$ knot. We show that for $n>2$ none of the knots in ${K^n_i}$ produces such counterexamples.

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Ryan Blair, Maggy Tomova. 2009-08-27. Companions of the unknot and width additivity. https://arxiv.org/abs/0908.4103

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