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The minimization of the number of colors is different at p = 11

Pedro Lopes

Center for Mathematical Analysis, Geometry, and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

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https://doi.org/10.1142/S0218216515500273Cited by:4 (Source: Crossref)

Abstract

In this article we present the following new fact for prime p = 11. For knots 6₂ and 7₂, mincol₁₁ 6₂ = 5 = mincol₁₁ 7₂, along with the following feature. There is a pair of diagrams, one for 6₂ and the other one for 7₂, each of them admitting only non-trivial 11-colorings using five colors, but neither of them admitting being colored with the sets of five colors that color the other one. This new fact 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 finding 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.

Keywords:

AMSC: 57M27

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