Examples on polynomial invariants of knots and links
type: Item
instance of: scholarly article
title: Examples on polynomial invariants of knots and links
zbMATH Open document ID: 0584.57005
DOI: 10.1007/BF01459137
author: Taizo Kanenobu
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164170
review text: After the discovery of the Jones polynomial [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New. Ser. 12, 103-111 (1985; Zbl 0564.57006)], other new polynomial invariants of knots and links have been discovered; the 2- variable Jones polynomial of \textit{P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, J. Millett, A. Ocneanu} [ibid. 12, 239-246 (1985; Zbl 0572.57002)] and \textit{J. H. Przytycki} and \textit{P. Traczyk} [to appear in Kobe J. Math.], which generalizes both the Jones polynomial and the reduced Alexander polynomial, and is a skein invariant, and the Q or absolute polynomial of R. D. Brandt, W. B. R. Lickorish, K. C. Millett, and C. F. Ho. We first give infinitely many skein equivalent knots which are classified by the Q polynomial. Second, we give an example of arbitrarily many prime knots with the same Jones but distinct Alexander polynomials, where the \(''V_{\infty}\) formula'' for the Jones polynomial of Jones and Birman is used. Lastly, we consider 2-bridge knots and links: We construct arbitrarily many 2-bridge knots with the same Jones and Q polynomials, and arbitrarily many skein equivalent fibered 2-bridge links with the same 2-variable Alexander and Q polynomials. Also we give a pair of skein equivalent 2-bridge links with the same Q polynomial, but distinct 2- variable Alexander polynomials. And we give an example of a fibered 2- bridge link which is transformed into a non-isotopic link by reversing the orientation of one component, but whose skein equivalent class and 2- variable Alexander polynomial do not change.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 3935986
zbMATH Keywords: Jones polynomial
zbMATH Keywords: polynomial invariants of knots
zbMATH Keywords: 2-variable Jones polynomial
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: skein invariant
zbMATH Keywords: skein equivalent knots
zbMATH Keywords: Q polynomial
zbMATH Keywords: 2-bridge knots
zbMATH Keywords: links
zbMATH Keywords: fibered 2-bridge links
MaRDI profile type: Publication
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OpenAlex ID: W3033577549
description: scientific article; zbMATH DE number 3935986