On the Minimum Number of Colors for Links: Change of Behavior at p=11
Pedro Lopes · arXiv (Cornell University) · 2013
This article concerns the minimum number of colors it takes to assemble a non-trivial coloring modulo a prime p,i.e. a non-trivial p-coloring. Let us use the term p-Minimal Sufficient Set of Colors for a set with the minimum number of colors, which realizes such a coloring. The work developed so far by several authors shows that for each prime p<11 there is a p-Minimal Sufficient Set of Colors which depends on the prime p and not on the link at issue. Namely for each such prime p there is a unique positive integer mp (the minimum number of colors modulo p), and a specific p-Minimal Sufficient Set of Colors, say {c1, c2, ..., cmp}, such that, given any link admitting non-trivial p-colorings, there is a diagram of the link equipped with a non-trivial p-coloring using colors c1, ..., cmp. In this article we show that this is no longer true for p=11. We also address obstructions to minimizing the number of colors and use this to calculate the minimum number of colors for a set of knots of determinant 11 and 13. Finally we propose directions for further work.