Equivalence classes of colorings

Volume 103 / 2014

Jun Ge, Slavik Jablan, Louis H. Kauffman, Pedro Lopes Banach Center Publications 103 (2014), 63-76 MSC: 57M27. DOI: 10.4064/bc103-0-2


For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial $m$-colorings of the link (an $m$-coloring has values in $\mathbf{Z}/m\mathbf{Z}$). Given a diagram of the link, the equivalence class of a non-trivial $m$-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).


  • Jun GeSchool of Mathematical Sciences
    Xiamen University
    Xiamen, Fujian 361005, P. R. China
  • Slavik JablanThe Mathematical Institute
    Knez Mihailova 36
    P.O. Box 367
    11001, Belgrade, Serbia
  • Louis H. KauffmanDepartment of Mathematics, Statistics and Computer Science
    University of Illinois at Chicago
    851 S. Morgan St.
    Chicago, IL 60607-7045, USA
  • Pedro LopesCenter for Mathematical Analysis, Geometry and Dynamical Systems
    Department of Mathematics
    Instituto Superior Técnico
    Universidade de Lisboa
    Av. Rovisco Pais
    1049-001 Lisbon, Portugal

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