Minimum Number of Fox Colors for Small Primes

Details Minimum Number of Fox Colors for Small Primes Minimum Number of Fox Colors for Small Primes

2010-01-08

arxiv.org/abs/1001.1334v2

Mathematics

Geometric Topology

12 pages, 2 figures, version accepted in JKTR

Scientific paper

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.

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