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In this paper, we show that any 11-colorable knot has a diagram with a non-trivial 11-coloring such that at most six colors among eleven colors are assigned to arcs of the diagram.
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... Theorem 4. Let p be an odd prime. Any p-colorable knot K satisfies mincol p (K) ≥ ⌊log 2 p⌋ + 2 (1) where ⌊x⌋ is the largest integer less than or equal to x. ...
... This equivalence relation among sets of colors unveils somewhat more the fascinating topic of Fox colorings. Clearly, mincol m L is an invariant of link L. For small primes, we have certain results for this invariant, listed as Theorem 5 [8,15,13,14,11,1]. However, the minimum number of colors is very difficult to compute in general, even for torus knots T (2, n) [8,9]. ...
... Oshiro [13] proved that any 7-colorable link with non-zero determinant can be colored by {0, 1, 2, 4}. And Cheng et al. [1] proved that any 11colorable link with non-zero determinant can be colored by 5 or 6 colors in {0, 1, 4, 6, 7, 8}. These three papers used similar techniques developed by Satoh. ...
In this paper we first investigate minimal sufficient sets of colors for p=11
and 13. For odd prime p and any p-colorable link L with non-zero determinant,
we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L
\geq 6 for p \geq 17. We 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. Finally, we
give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh
concerning an inequality involving crossing numbers. We show it is an equality
only for the trefoil and for the figure-eight knots.
... There is a number of articles on the topic of minimum number of colors [1,3,5,6,7,8,9,10,11,12,13]. In [13], Satoh developed a technique for finding the minimum number of colors over a fixed modulus but on an otherwise arbitrary situation. ...
... Then Oshiro [11] made the first impressive use of this technique by eliminating a string of 3 colors mod 7 thus showing that mod 7, 4 colors suffice. Using the same technique, Cheng et al [1] proved that at most 6 colors are needed mod 11, and Nakamura et al [10] proved further that 5 is the minimum number of colors for any knot or link admitting non-trivial 11-colorings. In the current article we apply Satoh's technique to prove the following result. ...
In this article we show that if a knot diagram admits a non-trivial coloring
modulo 13 then there is an equivalent diagram which can be colored with 5
colors. Thus the minimum number of colors modulo 13 is 5.
... For a p-colorable knot K, the number C p (K) is defined to be the minimum number of #Im(C) for all non-trivially p-colored diagrams (D, C) of K. This number has been studied in some papers [2,4,7,8,10,11,13,15,17]. In particular, it is shown in [11] that C p (K) ≥ log 2 p + 2 for any p-colorable knot K, and the equality holds for p = 3, 5, 7 [13,17]. ...
... For p = 11, we have C 11 (K) ≥ 5 by the above inequality or [10,Theorem 2.4]. On the other hand, it is proved in [2] that C 11 (K) ≤ 6. If an 11-colored diagram (D, C) satisfies #Im(C) = 5, then there are two possibilities Im(C) = {1, 4, 6, 7, 8}, {0, 4, 6, 7, 8} up to isomorphisms induced by affine maps of Z/11Z. ...
... In Section 3, we prove Theorem 1.1(i). The starting point of the proof is a modified version of the theorem in [2]: For any 11-colorable knot K, there is an 11-colored diagram (D, C) of K with Im(C) = {0, 1, 4, 6, 7, 8}. By applying Reidemeister moves to (D, C) suitably, we remove the color 0 from the diagram. ...
We prove that any $11$-colorable knot is presented by an $11$-colored diagram
where exactly five colors of eleven are assigned to the arcs. The number five
is the minimum for all non-trivially $11$-colored diagrams of the knot. We also
prove a similar result for any $11$-colorable ribbon $2$-knot.
This paper mainly studies the minimum number of colorings for all non-trivially 19-colored diagrams of any 19-colorable knot K. By using some special Reidemeister move, we successfully eliminated 13 colors from 19 colors. It can be seen that for any 19-colorable knot K, at least six colors are enough to color K, that is, the minimum number of 19-colorable knot is six.
We prove that for any odd (Formula presented.), the (Formula presented.)-palette number of any effectively (Formula presented.)-colorable (Formula presented.)-bridge knot is equal to (Formula presented.). Namely, there is an effectively (Formula presented.)-colored diagram of the (Formula presented.)-bridge knot such that the number of distinct colors that appeared in the diagram is exactly equal to (Formula presented.).
