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We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.
Contexts in source publication
Context 1
... D will necessarily have one of the two crossings, {2a + 1|a|16} or {a|16|15 − a} for some a = 16. Since 2a + 1 = 16 and 15 − a = 16, we deform the arc colored by a as shown in Figure 2 in the case of the first crossing, or as shown in Figure 3 in the case of the second crossing. Each of those two deformations provides an equivalent diagram where the crossing {16|16|16} disappeared. ...
Context 2
... of those two deformations provides an equivalent diagram where the crossing {16|16|16} disappeared. In the case of the second crossing, we do the deformation described in Figure 3. Case 2: Assume that D has a crossing whose overarc has the color 16, i.e. it is of the type {a|16|15 − a} for some a = 16. ...
Context 3
... the case of the second crossing, we do the deformation described in Figure 3. The obtained color 2a−c will be different from c and c k iff a = c and a = 9(c+c k ), for each k such that 1 ≤ k ≤ i − 1. ...
Context 4
... deformations. If a = b, we do the deformation described in Figure 6. We get the two new colors 2a − b and 2a − 2b + c. They are different from c and c k iff b = 2a − c, b = 2a − c k and b = a + 9c − 9c k , for each k, 1 ≤ k ≤ i − 1. Then the color c disappears and none of the colors c k appears. Step Fig. 42 (1, 4) (7, 9) (2, 14) (14, 2) Fig. 37 (7, 6) (1, 12) (12, 1) Fig. 43 7 (5, 16) (3, 1) (1, 3) Fig. 46 (5, 6) (11, 0) (0, 11) Fig. 45 (5, 7) (0, 12) (12, 0) ...
Context 5
... we do the deformation described in Figure 6. We get the two new colors 2a − b and 2a − 2b + c. They are different from c and c k iff b = 2a − c, b = 2a − c k and b = a + 9c − 9c k , for each k, 1 ≤ k ≤ i − 1. Then the color c disappears and none of the colors c k appears. Step Fig. 42 (1, 4) (7, 9) (2, 14) (14, 2) Fig. 37 (7, 6) (1, 12) (12, 1) Fig. 43 7 (5, 16) (3, 1) (1, 3) Fig. 46 (5, 6) (11, 0) (0, 11) Fig. 45 (5, 7) (0, 12) (12, 0) ...
Context 6
... 10, 16) (12, 14) (7,16,10) (14, 12) Fig. 36 7 (5, 10, 6) (3, 13) (5, 6, 10) (13, 3) Fig. 37 Tab. 5: ...
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Citations
... In knot theory, minimum numbers of colors for arc colorings have been studied in many papers (see [1,2,4,5,9,10,11] for example). We denote by mincol Fox p (K) the number for a Fox p-colorable knot K. ...
In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number p and any Dehn p-colorable knot K, the minimum number of colors for K is at least . Moreover, we will define the -palette graph for a set of colors. The -palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn p-colored diagram. In Appendix, we also prove that for Dehn 5-colorable knot, the minimum number of colors is 4.
... In knot theory, minimum numbers of colors of knots for Fox colorings are studied in many papers (see [1,2,3,4,5,8,9,11,13,15,16] for example). In particular, it is known that when p is an odd prime number, for any Fox p-colorable knot K, the minimum number of colors of K, denoted by mincol Fox p (K), satisfies that mincol Fox p (K) = ⌊log 2 p⌋ + 2 (see [5,8]). ...
... Considering how to connect each crossing, we can see that {1, 4}-semiarcs and {4, 7}-semiarcs make a knot component, and other arcs make other components (see Figure 10), which contradicts that K is a knot. Next, suppose that crossings of type (1,4), (1, 7) do not exist and crossings of type (0, 7), (0, 1) exist. Considering how to connect each crossing, we can see that {0, 7}-semiarcs and {1, 7}-semiarcs make a knot component, and other arcs make other components (see Figure 11), which contradicts that K is a knot. ...
... Suppose that crossings of type (1,8), (1,15) exist. Considering how to connect each crossings, we can see that {1, 8}-semiarcs and {8, 15}-semiarcs make a knot component, and other arcs make other components (see Figure 13), which contradicts that K is a knot. ...
In this paper, we give a method to evaluate minimum numbers of Dehn colors for knots by using symmetric local biquandle cocycle invariants. We give answers to some questions arising as a consequence of our previous paper [6]. In particular, we show that there exist knots which are distinguished by minimum numbers of Dehn colors.
... In 2017, Elhamdadi and Kerr [6] and independently Bento and Lopes [5] proved that C 13 (K) = 5. In 2020, Abchir, Elhamdadi and Lamsifer [1] showed that C 17 (K) = 6. In 2022, Han and Zhou [9] showed that C 19 (K) = 6. ...
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 ≥ 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.
... In 2017, Bento and Lopes [3] proved that C 13 (K) = 5. In 2020, Abchir et al. [1] proved that C 17 (K) = 6. ...
... The minimum number of coloring of knots Table 7. Elimination of color 1 from under-arcs joining crossings whose over-arcs bear the same color. Then none of the new arcs which are obtained by the above deformation have colors 1,5,6,8,9,11, 17 or 18. ...
... The minimum number of coloring of knots x = 7, 13, transform as in Fig. 4; in the second kind, the adjacent intersection is (x/0/19−x), so we need x, 19−x = 0, 1,4,5,6,8,9,11,15,16,17,18, that is, x = 7, 12. After the transformation as shown in Fig. 5, we need 2x = 0, 1, 4, 5, 6,8,9,11,15,16,17,18, that is, x = 12. ...
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.
This article serves two purposes. The first is to give an introduction to the readers who are not so familiar with quandle theory. The second is to report on new results. The first new result is that we compute second and third homology groups of disjoint union of quandles using the machinery of spectral sequences. We also establish a connection between graph theory and quandle theory via quandle rings. More specifically, we prove that for a finite commutative Latin quandle, the zero-divisor graph of the quandle ring with binary coefficients is connected and has a diameter bounded by two below and bounded by four above. We also investigate graph automorphisms of quandle rings and their relations with automorphisms of quandle rings and automorphisms of quandles. In particular we compute the ring automorphism groups of the quandle rings of the dihedral quandle of four elements and the trivial quandles of orders two, three and four.
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 Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered 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>=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 that one component.
