Jones polynomials and classical conjectures in knot theory
type: Item
instance of: scholarly article
title: Jones polynomials and classical conjectures in knot theory
zbMATH Open document ID: 0628.57004
DOI: 10.1016/0040-9383(87)90058-9
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: The paper contains proofs of several of the classical conjectures of P. G. Tait on knot projections. A regular projection \(\tilde L\) of a link \(L\subset S^ 3\) is called proper, if there are no evident ``flips'' reducing the number of double points. Theorem A: Two (connected and proper) alternating projections of an alternating link have the same number of double points. Theorem B: The minimal projection of an alternating link is alternating. In other words, an alternating link always has an alternating projection that has the minimum number of double points among all projections. Moreover, a nonalternating projection of a prime alternating link cannot be minimal. In addition, a corollary is proved: If a knot K is alternating and amphicheiral, then any proper alternating projection has an even number of double points. Furthermore, the twist number (writhe) is shown to be an invariant of proper special alternating projections. (The author and others have since shown that the restriction ``special'' can be dropped.) The proof establishes a close connection between the minimal number of double points of a projection \(\tilde L\) and the degree of Jones polynomial \(V_ L(t)\).
reviewed by: Gerhard Burde
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4022163
zbMATH Keywords: Tait conjecture
zbMATH Keywords: knot projections
zbMATH Keywords: alternating projections
zbMATH Keywords: alternating link
zbMATH Keywords: minimal projection
zbMATH Keywords: prime alternating link
zbMATH Keywords: amphicheiral
zbMATH Keywords: twist number
zbMATH Keywords: Jones polynomial
Wikidata QID: Q122919401
author: K. Murasugi
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90058-9
OpenAlex ID: W2017961228
description: scientific article; zbMATH DE number 4022163