Narrow Results

We resume work on telling embeddings of codimension two apart by counting colorings of the corresponding diagrams by given quandles. Previously, we illustrated the efficiency of this approach on classical knots. In the present paper we apply it to knotted surfaces. We recover work of Kamada in telling ribbon knots apart and we distinguish all elements of a class of twist-spun torus knots.

Colorings of torus knots and twist-spun torus knots by general Alexander quandles over finite fields are explicitly determined. It is then observed that, for some standard Mochizuki's cocycles, a large class of twist-spun torus knots with non-trivial colorings have trivial CJKLS state-sum invariants.

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.

In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

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 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.

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.