Minimum Number of Colors: the Turk’s Head Knots Case StudyReport as inadecuate

Minimum Number of Colors: the Turk’s Head Knots Case Study - Download this document for free, or read online. Document in PDF available to download.

1 Department of Mathematics Lisbonne 2 Center for Mathematical Analysis, Geometry and Dynamical Systems, Lisbon

Abstract : An $r$-coloring of a knot diagram is an assignment of integers modulo $r$ to the arcs of the diagram such that at each crossing, twice the the number assigned to the over-arc equals the sum of the numbers assigned to the under-arcs, modulo $r$. The number of $r$-colorings is a knot invariant i.e., for each knot, it does not depend on the diagram we are using for counting them. In this article we calculate the number of $r$-colorings for the so-called Turk-s Head Knots, for each modulus $r$. Furthermore, it is also known that whenever a knot admits an $r$-coloring using more than one color then all other diagrams of the same knot admit such $r$-colorings called non-trivial $r$-colorings. This leads to the question of what is the minimum number of colors it takes to assemble such an $r$-coloring for the knot at issue. In this article we also estimate and sometimes calculate exactly what is the minimum numbers of colors for each of the Turk-s Head Knots, for each relevant modulus $r$.

Keywords : Knots Turk’s head knots colorings colors minimum number of colors

Author: Pedro Lopes - João Matias -



Related documents