# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Banach Center Publications

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## Equivalence classes of colorings

### Tom 103 / 2014

Banach Center Publications 103 (2014), 63-76 MSC: 57M27. DOI: 10.4064/bc103-0-2

#### Streszczenie

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).

#### Autorzy

• Jun GeSchool of Mathematical Sciences
Xiamen University
Xiamen, Fujian 361005, P. R. China
e-mail
• Slavik JablanThe Mathematical Institute
Knez Mihailova 36
P.O. Box 367
e-mail
• Louis H. KauffmanDepartment of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 S. Morgan St.
Chicago, IL 60607-7045, USA
e-mail
• Pedro LopesCenter for Mathematical Analysis, Geometry and Dynamical Systems
Department of Mathematics
Instituto Superior Técnico