Mutation and volumes of knots in S 3
type: Item
instance of: scholarly article
title: Mutation and volumes of knots in S 3
zbMATH Open document ID: 0634.57005
DOI: 10.1007/BF01389038
author: Daniel Ruberman
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143507
review text: A mutation is an operation which associates a new knot K' to a knot \(K\subset S\) 3 as follows: Isotop the knot K so that it intersects transversely the unit 2-sphere S \(2\subset {\mathbb{R}}\) \(3=S\) 3-\(\infty\) in exactly the 4 points \((\pm 1,0,0)\), \((0,\pm 1,0)\); then K' is the union of K-B 3 and of R(K\(\cap B\) 3), where B 3 is the unit ball in \({\mathbb{R}}^ 3 \)and where R is the 180 o-rotation around the x-axis. The knots K and K' may be distinct, but it has been observed for a long time that many of their invariants are equal. For instance, they cannot be distinguished by their signature, their Alexander polynomial, their Jones polynomial, or the new 2-variable polynomial invariant generalizing these. The author goes one step further in this direction by proving that their complements S 3-K and S 3-K' have the same Gromov simplicial volume. In particular, if S 3-K admits a hypebolic metric (of finite volume) then S 3-K' admits a hyperbolic metric of the same volume. The proof is based on a cut-and- paste agument in hyperbolic geometry.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 4032621
zbMATH Keywords: mutation of knots
zbMATH Keywords: Gromov simplicial volume
zbMATH Keywords: hypebolic metric (of finite volume)
zbMATH Keywords: cut-and-paste
Wikidata QID: Q30054758
reviewed by: Francis Bonahon
MaRDI profile type: Publication
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OpenAlex ID: W2070143979
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description: scientific article; zbMATH DE number 4032621