https://hal.inria.fr/hal-01349054Lopes, PedroPedroLopesDepartment of Mathematics [Lisbonne] - IST - Instituto Superior Técnico, Universidade Técnica de LisboaCenter for Mathematical Analysis, Geometry and Dynamical Systems, Lisbon - IST - Instituto Superior Técnico, Universidade Técnica de LisboaMatias, JoãoJoãoMatiasDepartment of Mathematics [Lisbonne] - IST - Instituto Superior Técnico, Universidade Técnica de LisboaMinimum Number of Colors: the Turk’s Head Knots Case StudyHAL CCSD2015KnotsTurk’s head knotscoloringscolorsminimum number of colors[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, Coordination2016-07-26 17:19:052017-09-07 01:03:442016-07-26 17:56:37enJournal articleshttps://hal.inria.fr/hal-01349054/document10.46298/dmtcs.2139application/pdf1An $r$-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 $r$.