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Journal of Knot Theory and Its Ramifications
c
⃝World Scientific Publishing Company
Minimal Sufficient Sets of Colors and Minimum Number of Colors
Jun Ge
Department of Mathematics,
Sichuan Normal University,
Chengdu 610066, Sichuan, P. R. China
mathsgejun@163.com
Xian’an Jin
School of Mathematical Sciences,
Xiamen University,
Xiamen 361005, Fujian, P. R. China
xajin@xmu.edu.cn
Louis H. Kauffman
Department of Mathematics, Statistics and Computer Science,
University of Illinois at Chicago, 851 S. Morgan St.,
Chicago IL 60607-7045, USA
kauffman@uic.edu
Pedro Lopes
Center for Mathematical Analysis, Geometry, and Dynamical Systems,
Department of Mathematics,
Instituto Superior T´ecnico, Universidade de Lisboa
1049-001 Lisbon, Portugal
pelopes@math.tecnico.ulisboa.pt
Lianzhu Zhang
School of Mathematical Sciences,
Xiamen University,
Xiamen 361005, Fujian, P. R. China
zhanglz@xmu.edu.cn
ABSTRACT
In this paper we first investigate minimal sufficient sets of colors for p= 11 and 13.
For odd prime pand any p-colorable link Lwith det L̸= 0, we give alternative proofs of
mincolpL≥5 for p≥11 and mincolpL≥6 for p≥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.
Keywords: Link colorings; Minimum number of colors; Equivalence classes of colorings;
Minimal sufficient sets of colors.
Mathematics Subject Classification 2010: 57M27
1
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2J. Ge et al.
1. Introduction
A Fox m-coloring [5] is an assignment of elements from {0,1, . . . , m−1}to the arcs
of a link diagram such that at each crossing twice the integer assigned to the over-
arc equals to the sum of the integers assigned to the two under-arcs mod m. For
each link diagram and each modulus m > 1, there are always mtrivial colorings,
namely by assigning the same integer mod mto every arc of the diagram. A coloring
with at least two distinct colors (i.e., two distinct integers mod massigned to two
arcs) is called a non-trivial coloring. It is easy to check that if one diagram of a link
has a non-trivial m-coloring, then each diagram of it has a non-trivial m-coloring.
A link is called m-colorable if it admits a diagram with non-trivial m-colorings.
The following well-known theorem presents a criterion for checking if a given link
is m-colorable.
Theorem 1.1. A link Lis m-colorable if and only if the determinant of L(det L)
and mare not relatively prime.
The following concept was introduced by Harary and Kauffman in [7].
Definition 1.2. Given an integer mgreater than 1. Let Lbe a link admitting
non-trivial m-colorings. Let Da diagram of L, let nm,D be the minimum number
of colors mod mit takes to construct a non-trivial m-coloring on D. Set
mincolmL.
= min{nm,D |Dis a diagram of L}.
We call mincolmLthe minimum number of colors of L, mod m.
We call any non-trivial m-coloring of Lusing mincolmLcolors a minimal m-
coloring of L.
In this article we give a shorter proof of the following Theorem.
Theorem 1.3. [12] Let pbe a prime greater than 7. Let Lbe a link with p|det L̸=
0. Then mincolpL≥5.
We also prove the following fact.
Theorem 1.4. Let pbe a prime greater than 13. Let Lbe a link with p|det L̸= 0.
Then mincolpL≥6.
We give these proofs because we think they are both insightful and instructive.
In fact some of our methods lead to new research problems which we will detail
later in this paper.
Nakamura, Nakanishi, and Satoh proved a more general theorem in [13] for
knots, which we were unaware of at the time of the writing of our article using a
completely different approach.
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 3
Theorem 1.5. [13] Let pbe an odd prime. Any p-colorable knot Ksatisfies
mincolp(K)≥ ⌊log2p⌋+ 2 (1.1)
where ⌊x⌋is the largest integer less than or equal to x.
It is worth pointing out that Nakamura et al.’s proof for Theorem 1.5 can not
be naturally extended to p-colorable links with non-zero determinant. We would
like to understand whether links with non-zero determinant also have such a good
lower bound or not.
We also define an equivalence relation among sets of colors over a given modulus,
see Definition 2.8, and count the number of such equivalence classes for the least
mincolpfor p= 11, 13 and 17. This equivalence relation among sets of colors unveils
somewhat more the fascinating topic of Fox colorings.
2. Definitions and Preliminaries
Clearly, mincolmLis an invariant of link L. For small primes, we have certain results
for this invariant, listed as Theorem 2.1 [9,17,15,16,12,2]. However, the minimum
number of colors is very difficult to compute in general, even for torus knots T(2, n)
[9,10].
Theorem 2.1. Let Lbe a link with non-zero determinant.
(1) If 2|det L, then mincol2L= 2.
(2) If 3|det L, then mincol3L= 3.
(3) If 5|det L, then mincol5L= 4.
(4) If 7|det L, then mincol7L= 4.
(5) If pis a prime greater than 7and p|det L, then mincolpL≥5.
(6) If 11 |det L, then 5≤mincol11L≤6.
More precisely, Satoh [17] proved that any 5-colorable link with non-zero deter-
minant can be colored by {1,2,3,4}, mod 5. Oshiro [15] proved that any 7-colorable
link with non-zero determinant can be colored by {0, 1, 2, 4}, mod 7. Cheng et al.
[2] proved that any 11-colorable 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.
For p= 11, the fourth named author of this paper found an interesting be-
havior recently ([11]). In order to describe this discovery, we list some terminology
introduced in [11] first.
Definition 2.2. Let mbe a positive integer greater than 1 and let Lbe a link
admitting non-trivial m-colorings.
•An m-sufficient set of colors (for L) is a set of integers mod msuch that a
non-trivial m-coloring can be realized on a diagram of Lwith colors from this set.
•An m-minimal sufficient set of colors (for L) is an m-sufficient set of colors
(for L) whose cardinality is mincolmL.
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4J. Ge et al.
Definition 2.3. Suppose mis a positive integer greater than 1 such that for any
two links, Land L′, admitting non-trivial m-colorings, mincolmL= mincolmL′;
in this case, let mincol(m) denote this common minimum number of colors,
mincol(m) = mincolmL= mincolmL′.
If there is a set Sof mincol(m) elements from Zmsuch that for any m-colorable
link L′′, there exists a non-trivial m-coloring with colors from this set, then we call
Sa common m-minimal sufficient set of colors.
If there is a set S′of mincol(m) elements from Zmsuch that any diagram
supporting minimal m-colorings can be colored with colors from this set, then we
call S′a universal m-minimal sufficient set of colors.
Combining results proved in [9,17,15,16,12,11], it is proved in [11] that for each
prime p < 11, there is a universal p-minimal sufficient set of colors. Moreover, for
any of these primes, any common p-minimal sufficient set of colors is a universal
p-minimal sufficient set of colors.
Changes happen at p= 11. It is also proved in [11] that
Theorem 2.4. There is no universal 11-minimal sufficient set of colors.
It is worth to point out that it remains unknown whether there is a common
11-minimal sufficient set of colors or not.
To make our results easier to understand, we now introduce more terminology
and results already in the literature.
Definition 2.5. [6] Given a positive integer m, we define an m-coloring automor-
phism to be a permutation f, of the set Zm, such that
f(a∗b) = f(a)∗f(b)
for all a, b ∈Zm, with x∗y= 2y−x(mod m), for every x, y ∈Zm.
Proposition 2.6. m-coloring automorphisms preserve m-colorings.
Proposition 2.7. [4] Given a positive integer m, each m-coloring automorphism
is given by:
fλ,µ(x) = λx +µ
with µ∈Zmand λ∈Z∗
m, the set of units of Zm. Furthermore, the set of m-coloring
automorphisms equipped with composition of functions constitutes a group.
For a permutation fand a set {ai}i∈T, we denote f({ai}i∈T) = {f(ai)}i∈T. Let
Dbe an m-colorable diagram with arcs x1, . . . , xn. Suppose Cis an m-coloring of
Dwith arc xicolored by ai. Let fbe an m-coloring automorphism. Similarly, we
denote f(C) the m-coloring of Dsuch that arc xiis colored by f(ai).
Now it is natural for us to define an equivalence relation among color sets.
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 5
Definition 2.8. Given a positive integer m. Two color sets S1and S2are called
equivalent mod mif and only if there exists an m-coloring automorphism fλ,µ (x) =
λx +µ(mod m), such that f(S1) = S2.
Since coloring automorphisms preserve colorings, if a link Lcan be colored
with colors in a color set S, then it can be colored with colors in any color set
S′equivalent to S. Furthermore, if a color set Sis an m-sufficient set (m-minimal
sufficient set) for L, then any color set S′equivalent to Sis an m-sufficient set
(m-minimal sufficient set) for L.
Lemma 2.9. Let Lbe a non-split link. Let p≥3be a prime number. If a diagram D
of Ladmits a non-trivial p-coloring C, then there exist a p-coloring automorphism
fsuch that f(C)is a non-trivial p-coloring of Dcontaining colors 0, 1 and 2.
Proof. Since Lis a non-split link and pis an odd prime, Dmust have a non-
trivially colored crossing. Suppose at this crossing, the over-arc is colored by band
the two under-arcs are colored by aand c. Then 2b=a+c(mod p) and a, b, c are
distinct.
Let f(x) = (b−a)−1(x−a), where x−1is the inverse element of xin Zp. Then
f(a) = 0, f(b) = 1 and f(c) = c−a
b−a=2b−a−a
b−a= 2. Hence f(C) is a non-trivial
p-coloring of Dcontaining colors 0, 1 and 2.
The fourth author of this article studied what kind of color set would never be
a sufficient set or minimal sufficient set. The following two theorems are useful.
Theorem 2.10. [11] Let kbe a positive integer and La link with non-zero determi-
nant, admitting non-trivial (2k+ 1)-colorings. If S⊆ {0,1,2, . . . , k}, then Sis not
a(2k+ 1)-sufficient set of colors for L. Moreover, let fbe any (2k+ 1)-coloring au-
tomorphism. Then any set S′⊆ {f(0), f (1), ...f(k)}cannot be a (2k+ 1)-sufficient
set of colors for L.
Theorem 2.11. [11] Let kbe a positive integer and La link with non-zero de-
terminant, admitting non-trivial (2k+ 1)-colorings. Suppose S={c1, . . . , cn}is a
(2k+1)-sufficient set of colors for L. For each ci(i= 1,2, . . . , n), let Sibe the set of
unordered pairs {c1
i, c2
i}(with c1
i̸=c2
ifrom S) such that 2ci=c1
i+c2
imod 2k+ 1.
(1). There is at least one isuch that the expression 2ci=c1
i+c2
idoes not make
sense over the integers.
(2). If there is i∈ {1,2, . . . , n}such that for all j∈ {1,2, . . . , n}\{i},ciis not
in any of the unordered pairs in Sj, then n > mincol2k+1Land S\{ci}is also a
(2k+ 1)-sufficient set of colors for L.
Remark 2.12. Part (1) of Theorem 2.11 is a generalization of Theorem 2.10.
Motivated by Theorem 2.11, we define an associated edge-colored simple graph
for a color set.
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6J. Ge et al.
Definition 2.13. Let mbe a positive integer. For a mod mcolor set S=
{c1, c2, . . . , cn} ⊆ Zm, the associated edge-colored (or briefly, colored) graph Gc(S)
is a simple graph constructed as follows. Let {c1, c2, . . . , cn}be the vertex set of
Gc(S). There is a red edge between ciand cjif and only if there exists a positive
integer k≤n, such that 2ck=ci+cjover the integers. There is a blue edge between
ciand cjif and only if there exists an positive integer k≤n, such that 2ck=ci+cj
mod mbut 2ck̸=ci+cjover the integers. We denote G(S) the underlying graph
of Gc(S), i.e., the graph obtained by replacing each colored edge by a normal edge.
Remark 2.14. Nakamura et al. defined a “pallet graph” for a p-coloring in [13].
Their graph and ours are similar in some respects. But our definition focuses on
color sets while theirs focuses on colorings.
Now we rewrite and extend Theorem 2.11 by using the associated graph.
Theorem 2.15. Let kbe a positive integer and La link with non-zero determinant,
admitting non-trivial (2k+ 1)-colorings. Suppose S={c1, . . . , cn}is a (2k+ 1)-
sufficient set of colors for L. Let Gc(S)(G(S)) be the associated colored (underlying)
graph.
(1).Gc(S)contains at least one blue edge.
(2). If ciis an isolated vertex in G(S), then n > mincol2k+1Land S\{ci}is
also a (2k+ 1)-sufficient set of colors for L.
(3). Let c(G(S)) be the number of connected components of G(S). Let c(L)be
the component number of L. If c(G(S)) > c(L), then n > mincol2k+1Land there
exist c(L)components of G(S)such that the vertex set of these c(L)components is
also a (2k+ 1)-sufficient set of colors for L.
Corollary 2.16. (Corollary to both Theorem 2.11 and 2.15) {0,1,2,3,7}is not a
11-minimal sufficient set of colors for any link with non-zero determinant.
Proof. The graph associated with {0,1,2,3,7}is shown in Fig. 1. According to
part (1) of Theorem 2.11 or Theorem 2.15, {0,1,2,3,7}is not a 11-minimal sufficient
set of colors for any link with non-zero determinant.
Corollary 2.17. (Corollary to both Theorem 2.11 and 2.15) {0,1,2,4,10}is not
a 13-minimal sufficient set of colors for any link with non-zero determinant.
Proof. The graph associated with {0,1,2,4,10}is shown in Fig. 1. According to
either part (1) or part (2) of Theorem 2.11 or Theorem 2.15, {0,1,2,4,10}is not a
13-minimal sufficient set of colors for any link with non-zero determinant.
The following proposition is obvious and we omit its proof here.
Proposition 2.18. Let mbe a positive integer. Let S1and S2be two mod m
color sets with the same cardinality. If S1and S2are equivalent (mod m), then the
associated underlying graphs G(S1)and G(S2)are isomorphic.
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 7
Fig. 1. The left is the colored graph associated with {0,1,2,3,7}, the right is the colored graph
associated with {0,1,2,4,10}.
3. Why Links with non-zero determinant?
Most of our results in this paper concern links with non-zero determinant rather
than links or non-split links. It is because for links with zero determinant, there
exist “bad” examples when studying the minimum number of colors.
All split links (which have determinant zero) can be non-trivially colored with
two colors in any modulus. That is to say, for any split link and any modulus,
the minimum number of colors is 2. So Theorem 5.1, Theorem 5.2 and Theorem
1.5 cannot be extended to all links. But can we use non-split links instead of links
with non-zero determinant in these theorems? The following lemma gives a negative
answer.
Lemma 3.1. For any modulus m≥5, there are infinitely many non-split links
with zero determinant having minimum number of colors at most 4.
Proof. Figure 2 shows that for any modulus m≥5, the non-split link L8n8 in
the Thistlethwaite link table has a non-trivial coloring with color set {0,1,2,3}.
Note that L8n8 is (2,−2,2,−2)-pretzel link. It is easy to see that for any n,
(2,−2, . . . , 2,−2
2nstrands
)-pretzel link has the same property.
For pretzel link P(p1, p2, . . . , pn), the determinant is n
j=1
∏n
i=1 pi
pj[3]. Hence
det(P(2,−2, . . . , 2,−2)) = 0. This completes the proof.
4. Minimal Sufficient Sets of Colors
The fourth author of this article proved that unlike p≤7, there is no universal
11-minimal sufficient set of colors [11]. But it still remains unknown whether there
is a common 11-minimal sufficient set of colors or not. We now give further results
on 11-, 13-, and 17-sufficient or -minimal sufficient sets of colors.
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8J. Ge et al.
Fig. 2. A non-trivial coloring of L8n8 with 4 colors.
4.1. Minimal Sufficient Sets of Colors with Cardinality 5
Theorem 4.1. Let Lbe an 11-colorable link with non-zero determinant. If a di-
agram of Lcan be colored by a color set of 5 colors, then the color set must be
either {0,1,2,3,6}or {0,1,2,4,7}in the sense of equivalence of color sets induced
by coloring automorphism.
Proof. It was shown in [11] that, in the sense of equivalence class of color sets,
knot 62has unique 11-minimal sufficient set {0,1,2,3,6}and knot 72has unique 11-
minimal sufficient set {0,1,2,4,7}. It was also shown in [11] that {0,1,2,3,6}is not
equivalent to {0,1,2,4,7}. Recalling Lemma 2.9, we only need to consider color sets
of type {0,1,2, x, y}. Table 1 shows all instances of color sets of type {0,1,2, x, y}.
“type 1” (“type 2”) means it is equivalent to {0,1,2,3,6}({0,1,2,4,7}). “N, Th
2.10” (“N, Co 2.16”) means there is no link which can be 11-colored by it due to
Theorem 2.10 (Corollary 2.16). We pick three color sets as examples of how to read
Table 1.
(1) {0,1,2,3,8}. The coloring automorphism f10,3(x) = 10x+ 3 transforms the
color set {0,1,2,3,8}into {0,1,2,3,6}, so {0,1,2,3,8}is equivalent to {0,1,2,3,6}.
(2) {0,1,2,3,9}. The coloring automorphism f1,2(x) = x+ 2 transforms the
color set {0,1,2,3,9}into {0,2,3,4,5}, a subset of {0,1, . . . , 5}. According to The-
orem 2.10, {0,1,2,3,9}is not 11-sufficient set of colors for any link with non-zero
determinant.
(3) {0,1,2,4,9}. The coloring automorphism f6,1(x)=6x+ 1 transforms the
color set {0,1,2,4,9}into {0,1,2,3,7}. According to Corollary 2.16, {0,1,2,3,7}
is not a 11-minimal sufficient set of colors for any link with non-zero determinant.
Hence {0,1,2,4,9}is not a 11-minimal sufficient set of colors for any link with
non-zero determinant either.
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 9
Table 1. 5 color sets mod 11.
0,1,2,3,4 N, Th 2.10 0,1,2,5,7 (×2+1)=0,1,3,4,5 N, Th 2.10
0,1,2,3,5 N, Th 2.10 0,1,2,5,8 (×7)=0,1,2,3,7 N, Co 2.16
0,1,2,3,6 type 1 0,1,2,5,9 (×8+6)=0,1,2,3,6 type 1
0,1,2,3,7 N, Co 2.16 0,1,2,5,10 (+1)=0,1,2,3,6 type 1
0,1,2,3,8 (×10+3)=0,1,2,3,6 type 1 0,1,2,6,7 (×2)=0,1,2,3,4 N, Th 2.10
0,1,2,3,9 (+2)=0,2,3,4,5 N, Th 2.10 0,1,2,6,8 (×2)=0,1,2,4,5 N, Th 2.10
0,1,2,3,10 (+1)=0,1,2,3,4 N, Th 2.10 0,1,2,6,9 (×2)=0,1,2,4,7 type 2
0,1,2,4,5 N, Th 2.10 0,1,2,6,10 (+1)=0,1,2,3,7 N, Co 2.16
0,1,2,4,6 (×6)=0,1,2,3,6 type 1 0,1,2,7,8 (×2)=0,2,3,4,5 N, Th 2.10
0,1,2,4,7 type 2 0,1,2,7,9 (×5+1)=0,1,2,3,6 type 1
0,1,2,4,8 (×3)=0,1,2,3,6 type 1 0,1,2,7,10 (×10+2)=0,1,2,3,6 type 1
0,1,2,4,9 (×6+1)=0,1,2,3,7 N, Co 2.16 0,1,2,8,9 (+3)=0,1,3,4,5 N, Th 2.10
0,1,2,4,10 (+1)=0,1,2,3,5 N, Th 2.10 0,1,2,8,10 (+3)=0,2,3,4,5 N, Th 2.10
0,1,2,5,6 (×2+1)=0,1,2,3,5 N, Th 2.10 0,1,2,9,10 (+2)=0,1,2,3,4 N, Th 2.10
Theorem 4.2. Let Lbe a 13-colorable link with non-zero determinant. If a diagram
of Lcan be colored by 5 colors, then it can be only colored by {0,1,2,4,7}in
the sense of equivalence. Specifically, if mincol13L= 5 holds for all 13-colorable
links with non-zero determinant, then {0,1,2,4,7}is the only common 13-minimal
sufficient set of colors in the sense of equivalence of color sets induced by coloring
automorphism.
Proof. It was shown in [11] that, in the sense of equivalence class of color sets,
knots 63, 73and 10154 has unique 13-minimal sufficient set {0,1,2,4,7}. Recalling
Lemma 2.9, we only need to consider color sets of type {0,1,2, x, y}. Table 2 shows
circumstances of all color sets of type {0,1,2, x, y}. “Y” means it is equivalent to
{0,1,2,4,7}. “N, Th 2.10” (“N, Co 2.17”) means there is no link can be 11-colored
by it due to Theorem 2.10 (Corollary 2.17).
4.2. Possible Minimal Sufficient Sets of Colors with Cardinality 6
for p= 17
Now we determine which color sets with cardinality 6 may be minimal sufficient
sets of colors for primes 17. Our strategy is as follows.
Step 1. List all subsets of {0,1, . . . , p}with cardinality 6 and containing 0,1,
and 2.
Step 2. Classify these color sets into equivalence classes (recall Definition 2.8).
Step 3. Use Theorem 2.10 and Theorem 2.11 to check the color sets in order.
If Scan not be a p-minimal sufficient set of colors for any link with non-zero
determinant, then delete all those color sets equivalent to S. We call the remaining
ones possible p-sufficient sets of colors with cardinality 6.
Our results follow. We use a C program to achieve Step 1 and 2 (see Appendix
A).
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10 J. Ge et al.
Table 2. 5 color sets mod 13.
0,1,2,3,4 N, Th 2.10 0,1,2,5,12 (+1)=0,1,2,3,6 N, Th 2.10
0,1,2,3,5 N, Th 2.10 0,1,2,6,7 (×2+1)=0,1,2,3,5 N, Th 2.10
0,1,2,3,6 N, Th 2.10 0,1,2,6,8 (×2+1)=0,1,3,4,5 N, Th 2.10
0,1,2,3,7 (×2)=0,1,2,4,6 N, Th 2.10 0,1,2,6,9 (×3)=0,1,3,5,6 N, Th 2.10
0,1,2,3,8 (×2)=0,2,3,4,6 N, Th 2.10 0,1,2,6,10 (×3)=0,3,4,5,6 N, Th 2.10
0,1,2,3,9 (×2)=0,2,4,5,6 N, Th 2.10 0,1,2,6,11 (×10+7)=0,1,2,4,7 Y
0,1,2,3,10 (+3)=0,3,4,5,6 N, Th 2.10 0,1,2,6,12 (×2+2)=0,1,2,4,6 N, Th 2.10
0,1,2,3,11 (+2)=0,2,3,4,5 N, Th 2.10 0,1,2,7,8 (×2)=0,1,2,3,4 N, Th 2.10
0,1,2,3,12 (+1)=0,1,2,3,4 N, Th 2.10 0,1,2,7,9 (×2)=0,1,2,4,5 N, Th 2.10
0,1,2,4,5 N, Th 2.10 0,1,2,7,10 (×2)=0,1,2,4,7 Y
0,1,2,4,6 N, Th 2.10 0,1,2,7,11 (×6+1)=0,1,2,4,7 Y
0,1,2,4,7 Y 0,1,2,7,12 (×2+2)=0,2,3,4,6 N, Th 2.10
0,1,2,4,8 (×7)=0,1,2,4,7 Y 0,1,2,8,9 (×2)=0,2,3,4,5 N, Th 2.10
0,1,2,4,9 (×3+1)=0,1,2,4,7 Y 0,1,2,8,10 (×11+4)=0,1,2,4,10 N, Co 2.17
0,1,2,4,10 N, Co 2.17 0,1,2,8,11 (×12+2)=0,1,2,4,7 Y
0,1,2,4,11 (+2)=0,2,3,4,6 N, Th 2.10 0,1,2,8,12 (×2+2)=0,2,4,5,6 N, Th 2.10
0,1,2,4,12 (+1)=0,1,2,3,5 N, Th 2.10 0,1,2,9,10 (+4)=0,1,4,5,5 N, Th 2.10
0,1,2,5,6 N, Th 2.10 0,1,2,9,11 (+4)=0,2,4,5,6 N, Th 2.10
0,1,2,5,7 (×2)=0,1,2,4,10 N, Co 2.17 0,1,2,9,12 (+4)=0,3,4,5,6 N, Th 2.10
0,1,2,5,8 (×11+4)=0,1,2,4,7 Y 0,1,2,10,11 (+3)=0,1,3,4,5 N, Th 2.10
0,1,2,5,9 (×3)=0,1,2,3,6 N, Th 2.10 0,1,2,10,12 (+3)=0,2,3,4,5 N, Th 2.10
0,1,2,5,10 (×3)=0,2,3,4,6 N, Th 2.10 0,1,2,11,12 (+2)=0,1,2,3,4 N, Th 2.10
0,1,2,5,11 (×12+2)=0,1,2,4,10 N, Co 2.17
For p= 17, there are 14
3= 364 color sets with cardinality 6 and containing
0,1, and 2. They can be classified into 49 equivalence classes. Among them, there
are 9 possible 17-minimal sufficient sets of colors up to the equivalence relation:
{0,1,2,3,5,9},{0,1,2,3,5,10},{0,1,2,3,5,12},{0,1,2,3,6,9},{0,1,2,3,6,10},
{0,1,2,3,6,11},{0,1,2,3,6,13},{0,1,2,3,6,14}, and {0,1,2,3,7,10}.
5. Minimum Number of Colors
In this section, we study the lower bound of the minimum number of colors. First
we give a short proof of mincolpL≥5 for any p-colorable link Lwith det L̸= 0 and
prime p≥11. Then we show that we can go further by using a similar approach.
Theorem 5.1. [12] Let Lbe a link with non-zero determinant. If there is a prime
p≥11 such that Ladmits non-trivial p-colorings, then mincolpL≥5.
Proof. It is easy to prove the mod pminimum number of colors is at least 4 for
any link with non-zero determinant and any prime p≥5. See Lemma 2.1 in [17] for
example. So we only need to prove Lcannot be colored by 4 colors. Recalling Lemma
2.9, it is enough to consider color set Sof type {0,1,2, a}, where 3 ≤a≤p−1. If
Lcan be colored by S, then p+1
2≤a≤p−1 due to Theorem 2.10. The coloring
automorphism f(x) = x+ (p−a) transforms Sinto {0, p −a, p −a+ 1, p −a+ 2}.
Hence p−a+ 2 ≥p−1
2+ 1 due to Theorem 2.10, thus a≤p+3
2. So a∈ {p+1
2,p+3
2}.
We use the same notations as in Theorem 2.11. Let c1= 0, c2= 1, c3= 2, c4=a.
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 11
Since a∈ {p+1
2,p+3
2}, we have 6 ≤p+1
2≤c4≤p+3
2=p−p−3
2≤p−4. It is easy to
check S1=∅,S2={{0,2}} and S3=∅. So c4is not in any of the unordered pairs
in Sj,j∈ {1,2,3}. According to part (2) of Theorem 2.11, 4 >mincolpL, which
never happens.
Theorem 5.2. Let Lbe a link with non-zero determinant. If there is a prime
p≥17 such that Ladmits non-trivial p-colorings, then mincolpL≥6.
Proof. Recalling part (5) of Theorem 2.1, we only need to prove Lcannot be
colored by 5 colors. Suppose Lcan be colored by a color set Swith |S|= 5. Then
Sis a p-minimal sufficient set of colors for L.
Claim. Suppose S′={x1, x2, x3, x4} ⊆ Swith x1< x2< x3< x4. Then
x4−x1≥5.
Proof of the Claim. We prove the Claim by contradiction. If there is an S′
such that x4−x1≤4, then the coloring automorphism f1,p−x1(x) = x+p−x1
transforms Sinto a set S′′ ={0, y1, y2, y3, y4}, where y1< y2< y3≤4 and
y3< y4. According to Theorem 2.10, y4≥p+1
2and p−y4+ 4 ≥p+1
2, which yields
9≤y4≤p−5. So y4is not in any of the unordered pairs in any Sj(which means
y4cannot be the color of any under-arc at a polychromatic crossing). Recalling
Theorem 2.11, S′′ (so that S) is not a p-minimal sufficient set of colors for Land
mincolpL≤4, which is impossible. The proof is complete.
Recalling Lemma 2.9, it is enough to consider color set Sof type {0,1,2, a, b},
where 3 ≤a≤b≤p−1. Then p+1
2≤b≤p−1 due to Theorem 2.10.
The Claim indicates that a≥5 immediately. It also indicates that b≤p−3,
otherwise the coloring automorphism f1,p−b(x) = x+p−bwill transform {0,1,2, b}
into {0, p−b, p−b+1, p −b+2}where p−b+2 ≤4. Hence a≥5 and p+1
2≤b≤p−3.
So there is no crossing with the over-arc colored by one color from {0,1,2}, whose
under-arcs are colored with one color from {0,1,2}, and the other from {a, b}.
We keep all the notations used in Theorem 2.11. Let c1= 0, c2= 1, c3= 2,
c4=a,c5=b. Now we divide all the possibilities into the following two cases.
Case 1. a+b /∈ {0,2,4}mod p.
In this case, both aand bare not in any of the unordered pairs in S1,S2and
S3. So amust be in one of the unordered pairs in S5and bmust be in one of the
unordered pairs in S4according to Theorem 2.11, i.e., ∃α, β ∈ {0,1,2}(αand βare
not necessarily different), such that
2b=a+α+p
2a=b+β.(5.1)
Hence
a=p+α+2β
3
b=2p+2α+β
3
.(5.2)
The system of equations (5.1) is over integers, so are (5.3) and (5.5) below.
Since α, β ∈ {0,1,2}, the coloring automorphism f3,0(x) = 3xtransforms Sinto
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12 J. Ge et al.
{0,3,6, α + 2β, 2α+β} ⊆ {0,1, . . . , p−1
2}. So Sis not a p-sufficient set of colors for
L.
Case 2. a+b=γ∈ {0,2,4}mod p.
It is easy to see c2= 1 is not in any of the unordered pairs in S1and S3. So
1 must be in one of the unordered pairs in S4or S5, i.e., either 2a= 1 + bor
2b= 1 + a+p(over integers). So there are two subcases.
Case 2.1. 2a= 1 + b.
In this subcase,
2a= 1 + b
a+b=γ+p.(5.3)
Hence
a=p+γ+1
3
b=2p+2γ−1
3
.(5.4)
Since γ∈ {0,2,4}, the coloring automorphism f3,1(x) = 3x+ 1 transforms Sinto
{1,4,7, γ + 2,2γ} ⊆ {0,1, . . . , p−1
2}. So Sis not a p-sufficient set of colors for L.
Case 2.2. 2b= 1 + a+p.
In this subcase,
2b= 1 + a+p
a+b=γ+p.(5.5)
Hence
a=p+2γ−1
3
b=2p+γ+1
3
.(5.6)
Since γ∈ {0,2,4}, the coloring automorphism f3,1(x) = 3x+ 1 transforms Sinto
{1,4,7,2γ, γ + 2} ⊆ {0,1, . . . , p−1
2}. So Sis not a p-sufficient set of colors for L.
Therefore, Sis not a p-minimal sufficient set of colors for L, a contradiction.
6. Recent Progresses
After our early version of this paper was finished, some nice results on minimal
number of colors was achieved by now. Ichihara and Matsudo [8] answered a ques-
tion of ours raised in Section 1, extending Theorem 1.5 to effectively n-colored links
with non-zero determinant. Using results in the present paper, Nakamura, Nakan-
ishi and Satoh [14] proved that mincol11L= 5 and Bento and the fourth named
author of this paper [1] proved mincol13L= 5 for links with non-zero determinant.
Appendix A. C Program Used In Sec. 4
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 13
#include<stdio.h>
#include<stdlib.h>
#define MOD 17
#define CNT ((MOD - 3) * (MOD - 4) * (MOD - 5) / 6)
int intcomp(const int* a1, const int* a2)
{
return *(int*)a1 - *(int*)a2;
}
int array_same(int* a1, int* a2)
{
int aa1[6], aa2[6];
int i = 0;
for(i = 0; i < 6; i++)
{
aa1[i] = a1[i];
aa2[i] = a2[i];
}
qsort(aa1, 6, sizeof(int), intcomp);
qsort(aa2, 6, sizeof(int), intcomp);
for(i = 0; i < 6; i++)
if(aa1[i] != aa2[i])
return 0;
return 1;
}
int array_same_class(int* a1, int* a2, int* xx, int* yy)
{
int x, y, i, a3[6];
for(x = 1; x < MOD; x++)
for(y = 0; y < MOD; y++)
{
for(i = 0; i < 6; i++)
a3[i] = (a2[i] * x + y) % MOD;
if(array_same(a1, a3))
{
*xx = x; *yy = y;
return 1;
}
March 6, 2016 6:10 WSPC/INSTRUCTION FILE color
14 J. Ge et al.
}
return 0;
}
int flag[CNT] = {0};
int matrix[CNT][6];
void fill_matrix()
{
int i = 0;
int a, b, c;
for(a = 3; a < MOD; a++)
for(b = a+1; b < MOD; b++)
for(c = b+1; c < MOD; c++)
{
matrix[i][0] = 0;
matrix[i][1] = 1;
matrix[i][2] = 2;
matrix[i][3] = a;
matrix[i][4] = b;
matrix[i][5] = c;
i++;
}
}
void print_arr(FILE* pf, int* arr)
{
fprintf(pf, "%d %d %d %d %d %d\n",
arr[0],arr[1],arr[2],arr[3],arr[4],arr[5]);
}
int main()
{
int i, j, x, y;
FILE* pf = fopen("result.txt", "w");
fill_matrix();
for(i = 0; i < CNT; i++)
{
if(flag[i]) continue;
print_arr(pf, matrix[i]);
for(j = i + 1; j < CNT; j++)
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Minimal Sufficient Sets of Colors and Minimum Number of Colors 15
if(array_same_class(matrix[i], matrix[j], &x, &y))
{
print_arr(pf, matrix[j]);
// fprintf(pf, "%d %d \n", x, y);
flag[j] = 1;
}
fprintf(pf, "\n");
}
fclose(pf);
return 0;
}
March 6, 2016 6:10 WSPC/INSTRUCTION FILE color
16 J. Ge et al.
Acknowledgments
X. Jin and J. Ge acknowledge support from the National Natural Science Foun-
dation of China (No. 11271307). P. Lopes acknowledges partial funding from
FCT (Portugal) through projects PEst-OE/EEI/LA0009/2013, and EXCL/MAT-
GEO/0222/2012 (“Geometry and Mathematical Physics”). L. Zhang acknowledges
support from the National Natural Science Foundation of China (Nos. 11171279
and 11471273). The authors thank Prof. Satoh for kindly sending us [13]. J. Ge
thanks Jingkang Zhou for helping writing the program.
References
[1] F. Bento, P. Lopes, The minimum number of Fox colors modulo 13 is 5, preprint
(2015), arXiv:1508.07559.
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