Companions of the unknot and width additivity

Ryan Blair; Maggy Tomova

Companions of the unknot and width additivity

Geometric Topology 2009-08-31 v1

Authors: Ryan Blair , Maggy Tomova

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.

Keywords

unknot companions knot width additivity conjecture bridge number knot connected sum Scharlemann-Thompson knots

Cite

@article{arxiv.0908.4103,
  title  = {Companions of the unknot and width additivity},
  author = {Ryan Blair and Maggy Tomova},
  journal= {arXiv preprint arXiv:0908.4103},
  year   = {2009}
}

Comments

12 pages, 11 figures

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