Equivalence classes of colorings
Volume 103 / 2014
Banach Center Publications 103 (2014), 63-76
MSC: 57M27.
DOI: 10.4064/bc103-0-2
Abstract
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).
