Minimum Number of Fox Colors for Small Primes

TL;DR

This paper determines the exact minimum number of colors needed for Fox colorings of links with non-null determinant over various moduli, revealing specific values for small prime divisors and proposing a conjecture for all primes.

Contribution

It provides exact results for the minimum number of colors in Fox colorings for links with certain prime divisors of their determinants and introduces a conjecture generalizing these findings.

Findings

Minimum colors for prime divisors 2, 3, 5, 7 are 2, 3, 4, 4 respectively.

Establishes a pattern linking prime divisors to minimum coloring numbers.

Proposes a conjecture for the uniqueness of the minimum number of colors for all primes.

Abstract

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely. We are thus led to conjecture that for each prime p there exists a unique positive integer, m, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is m.