Unknotting number, genus and companion tori
type: Item
instance of: scholarly article
title: Unknotting number, genus and companion tori
zbMATH Open document ID: 0616.57003
DOI: 10.1007/BF01456051
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164360
review text: We apply the Gabai sutured manifold machine to study the effect of crossing changes on the genus of a knot. Our results are of two types: First, we show that there are knots for which a crossing change which lowers the unknotting number raises the genus. In fact the argument seems, in principle, to produce a myriad of such examples. Second, we examine the effect of companion tori on crossing changes. We discover that either such companion tori are essentially unrelated to the part of the knot on which the crossing change is being made, or there is an inequality which relates the genus of the knot before and after the crossing change. The precise statement and some corollaries are: Definition: Let K be a knot in \(S^ 3\) and D be a disk which intersects K in precisely two points, of opposite orientation. Let \(L=\partial D\) be given a \(\pm 1\)-framing. Then L is a crossing link and D is its crossing disk. Denote by K' the knot in \(S^ 3\) obtained from K by doing \(\pm 1\) surgery to L. (K and K' are related by a single crossing change.) Let T be an incompressible torus in \(S^ 3-K.\) Theorem: If genus(K')\(\leq genus(K)-2\) then any companion torus of K can be isotoped disjoint from L. Corollary: If K is a satellite knot of winding number \(\neq 0\), then the unknotting number of K is greater than one. Corollary: An unknotting number one knot is prime.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 3998780
zbMATH Keywords: sutured manifold
zbMATH Keywords: crossing changes
zbMATH Keywords: genus of a knot
zbMATH Keywords: unknotting number
zbMATH Keywords: companion tori
zbMATH Keywords: crossing link
zbMATH Keywords: crossing disk
zbMATH Keywords: incompressible torus
zbMATH Keywords: satellite knot
zbMATH Keywords: winding number
author: Martin Scharlemann
author: Abigail Thompson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3998780