The Minimization of the Number of Colors is Different at p=11

TL;DR

This paper investigates the minimal number of colors needed for non-trivial colorings of knots at prime p=11, revealing unique behaviors and establishing minimal color counts for certain knots, contrasting with smaller primes.

Contribution

It presents new findings on color minimization at p=11, including specific examples and obstructions, and extends results to knots with prime determinants 11 and 13.

Findings

For knots 6_2 and 7_2, mincol_{11} = 5.

There exist pairs of diagrams with identical minimal color sets that do not overlap.

The minimal number of colors is 5 for knots with prime determinants 11 and 13.

Abstract

In this article we present the following new fact for prime p=11. For knots 6_2 and 7_2, mincol_{11} 6_2 = 5 = mincol_{11} 7_2, along with the following feature. There is a pair of diagrams, one for 6_2 and the other one for 7_2, each of them admitting only non-trivial 11-colorings using 5 colors, but neither of them admitting being colored with the sets of 5 colors that color the other one. This is in full contrast with the behavior exhibited by links admitting non-trivial p-colorings over the smaller primes, p=2, 3, 5 or 7. We also prove results concerning obstructions to the minimization of colors over generic odd moduli. We apply these to find the right colors to eliminate from non-trivial colorings. We thus prove that 5 is the minimum number of colors for each knot of prime determinant 11 or 13 from Rolfsen's table.