Coloring link diagrams with a continuous palette
type: Item
instance of: scholarly article
title: Coloring link diagrams with a continuous palette
zbMATH Open document ID: 0964.57008
DOI: 10.1016/S0040-9383(99)00025-7
author: Daniel S. Silver
author: Susan G. Williams
published in: Topology
publication date: 2001-07-13T00:00:00Z
review text: A \(3\)-coloring of a link diagram, which was introduced by Fox, is an assignment of one of three possible colors to each arc in such a way that at each crossing either one or three colors are used. Generalizing the palette, more general notions have been introduced. For example, an \(n\)-coloring of a link diagram is an assignment of colors to each arc in such a way that at each crossing the sum of colors of the undercrossings is equal to twice the color of the overcrossing modulo \(n,\) where the palette is identified with the finite cyclic group \(Z_n.\) The number of different \(n\)-colorings of the diagram is an invariant of the link which is related to the Jones polynomial, the unknotting number, statistical mechanics, etc. In particular, the set of \(n\)-colorings is an abelian group that is isomorphic to \(H_1(M_2(\ell); Z_n) \oplus Z_n,\) where \(M_r(\ell)\) denotes the \(r\)-fold cyclic cover of \(S^3\) branched over \(\ell.\) Additional details may be found in [\textit{J. H. Przytycki}, Banach Cent. Publ. 42, 275-295 (1998; Zbl 0904.57002)]. More recently, the idea of coloring the arcs of a link diagram is used to construct more general notions [\textit{L. H. Kauffman}, Knots and physics, Series on Knots and Everything, 1. Singapore etc.: World Scientific (1991; Zbl 0733.57004)]. In the paper under review the authors consider a continuous palette \(T^Z\) consisting of all bi-infinite sequences \((\alpha_j)\) of elements \(\alpha_j \in T = R/Z\) instead of a finite one and show that this yields some significant results for the topology of the link \(\ell\) itself and the \(r\)-fold cyclic cover \(M_r(\ell)\) of \(S^3\) branched over \(\ell,\) and also asymptotic results about the homology of \(M_r(\ell_s)\) for the periodic braids \(\alpha^s\) for any positive integer \(s.\) More precisely, a \(T^Z\)-coloring of a diagram \(D\) of an oriented link \(\ell\) is defined to be an assignment of elements \(C \in T^Z\) to the arcs of \(D\) such that \(\sigma(C_i …
reviewed by: Sang Youl Lee
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1526238
zbMATH Keywords: \(n\)-coloring
zbMATH Keywords: branched cyclic cover of a link
zbMATH Keywords: braid entropy
MaRDI profile type: Publication
full work available at URL: s0040-9383(99)00025-7
OpenAlex ID: W2060896651
Wikidata QID: Q127814057
description: scientific article; zbMATH DE number 1526238