Narrow Results

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.

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 knot of type (2, p), modulo p. This expression is t + 2l -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 the 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.

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of colors, mod p, is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove that there exists a non-trivial coloring of any Schubert Normal Form (SNF) of it, modulo its determinant, which uses all colors available. If this determinant is an odd prime, then any non-trivial coloring of this SNF, modulo the determinant, uses all available colors. We prove also that the number of crossings in the SNF equals twice the determinant of the link minus 2. Facts about torus links and their coloring abilities are also proved.

In this paper, we first investigate the minimal sufficient sets of colors for $p=11$ and $13$. For odd prime $p$ and any $p$-colorable link $L$ with $detL\ne 0$, we give alternative proofs of ${\mathrm{\text{mincol}}}_{p}L\ge 5$ for $p\ge 11$ and ${\mathrm{\text{mincol}}}_{p}L\ge 6$ for $p\ge 17$. We also elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo $11$, and only one such class of five colors modulo 13.

This article concerns Fox colorings. We prove that if a link admits non-trivial $(2k+1)$-colorings, with prime $2k+1\ge 11$, it also admits non-trivial $(2k+1)$-colorings not involving colors $2k,2k-1$, nor $k$.

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree $k$ of the reduced Alexander polynomial of the knot. We show that it is exactly $k+1$ for L-space knots. Then we apply these results to torus knots and Pretzel knots $P(-2,3,2l+1)$, $l\ge 0$. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more than one component.