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MINIMUM NUMBER OF FOX COLORS FOR SMALL PRIMES

Center for Mathematical Analysis, Geometry and Dynamical Systems, Portugal

Department of Mathematics, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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JOÃO MATIAS

Department of Mathematics, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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https://doi.org/10.1142/S0218216511009728Cited by:12 (Source: Crossref)

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_p, 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_p.

Keywords:

AMSC: 57M27

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