The Teneva Game

TL;DR

This paper derives an upper bound for the minimum colors needed to non-trivially color certain torus knots modulo a prime p, showing the ratio decreases as p increases, and establishes minimal colors for specific knots.

Contribution

It introduces a formula for the upper bound on colors needed for T(2, p) knots using Teneva transformations and analyzes the ratio of required colors to available colors.

Findings

Upper bound formula for coloring T(2, p) knots.

Ratio of needed colors to available colors decreases with larger p.

Minimum colors for T(2, 11) is 5.

Abstract

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.