Coloring link diagrams with a continuous palette
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 \(\sigma(C_i - C_k) = C_j - C_k\) is satisfied at any crossing, where \(\sigma : T^Z \to T^Z\) is the shift automorphism which sends \((\alpha_j)\) to \((\alpha_{j+1})\) for \((\alpha_j) \in T^Z\), \(C_k\) corresponds to the overcrossing, \(C_i, C_j\) correspond to the undercrossings, and one encounters \(C_j\) as one travels in the preferred direction along the arc labelled by \(C_k,\) turning left at the crossing. Then the authors prove that the set of all \(T^Z\)-colorings of \(D\) forms a group characterized by the group of homomorphisms from the augmentation subgroup of the link group to \(T\) and that it is an invariant of the oriented link \(\ell\) which contains the group of all \(n\)-colorings of \(D.\) For a positive integer \(r\), a periodic \(T^Z\)-coloring with period \(r\) is a \(T^Z\)-coloring of a diagram \(D\) satisfying \(\sigma^r(C) = C\) for every assigned label \(C.\) Then a \((T, r)\)-coloring of \(D\) is a periodic \(T^Z\)-coloring with period \(r\) such that the sum of any \(r\) consecutive coordinates of any assigned label vanishes. The authors prove that the set of \((T, r)\)-colorings of \(D\) is an abelian group and its quotient by the connected component of the identity is isomorphic to the torsion subgroup \(TH_1(M_r(\ell); Z)\) of the homology group \(H_1(M_r(\ell); Z).\) On the other hand, the authors determine the exponential growth rates \(\lim_{s \to \infty}(1/s)\log |TH_1(M_r(\ell_s); Z)|\) for each \(r,\) where \(\ell_s\) denotes the closure of a periodic braid \(\alpha^s.\) Using this they obtain that if the exponential growth rate of \(TH_1(M_r(\ell); Z)\) is positive for some integer \(r,\) then the braid \(\alpha\) has positive braid entropy. In the last section, the authors observe that their approach can be generalized with \(T\) replaced by an arbitrary topological group.
scientific article; zbMATH DE number 1526238