Minimum Number of Colors: the Turk’s Head Knots Case Study
Pedro Lopes, Joao Matias
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.
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