A criterion for knots of period 3
type: Item
instance of: scholarly article
title: A criterion for knots of period 3
zbMATH Open document ID: 0706.57001
DOI: 10.1016/0166-8641(90)90051-3
author: Paweł Traczyk
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The criterion uses the two-variable polynomial P(m,\(\ell)\); \(P_ 0\) denotes its terms of degree zero in m. Theorem 1. Assume a knot K has period 3. Let \(P_ 0(K)=\sum a_ k\ell^{2k}\). Then for every k, \(a_{3k+1}+a_{3k+2}\equiv 0 mod 3.\) The criterion can be successfully applied to a number of knots whose period 3 was doubtful. The proof reduces the 3-periodic knot diagram to that of a 3-periodic closed braid and finally to that of a torus knot t(k,3). A direct computation concludes the proof using a formula of V. Jones.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4159634
zbMATH Keywords: homfly polynomial
zbMATH Keywords: two-variable polynomial
zbMATH Keywords: 3-periodic knot diagram
zbMATH Keywords: 3- periodic closed braid
zbMATH Keywords: torus knot
reviewed by: Gerhard Burde
MaRDI profile type: Publication
full work available at URL: 0166-8641(90)90051-3
OpenAlex ID: W1983564577
cites work: Knots
cites work: Hecke algebra representations of braid groups and link polynomials
cites work: Signatures of Covering Links
cites work: A polynomial invariant of oriented links
cites work: Jones polynomials of periodic links
cites work: On Murasugi's and Traczyk's criteria for periodic links
cites work: 10 101 Has no Period 7: A Criterion for Periodic Links
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description: scientific article; zbMATH DE number 4159634