11

Minimal coloring numbers of \mathb {Z} ‐colorable links
Kazuhiro Ichihara and Eri Matsudo

College of Humanities and Sciences, Nihon University
1

Introduction

The aim of this article is to give a summery of the results given in the papers [5, 8, 6] on\mathb {Z} ‐colorings of links.

In [3], Fox introduced one of the most well‐known invariants for knots and links in the
3‐space, now called the Fox n ‐coloring, or simply n‐coloring for n\geq 2 . For example, thetricolorability is much often used to prove that the trefoil is a non‐trivial knot. However,some of links are known to admit non‐trivial n‐colorings for any n\geq 2 . In fact, if thedeterminant of a link L is 0 , then it is shown that L admits non‐trivial n‐coloring for

any n\geq 2 . (For example, see [5] for the definition of the determinant of a link.) In that}case, L admits a generalization of the Fox n‐coloring, which we call a \mathbb{Z‐‐coloring
definedas follows.

Definition 1. Let L be a link and D a regular diagram of L . We consider a map \gamma : {arcof D\}arrow \mathbb{Z}. If \gamma satisfies the condition 2\gamma(a)=\gamma(b)+\gamma(c) at each crossing of D with theover arc a and the under arcs b and c , then \gamma is called a \mathb {Z} ‐coloring on D. A\mathbb{Z} ‐coloring
which assigns the same value to all the arcs of the diagram is called a trivial \mathb {Z} ‐coloring.
A link is called \mathb {Z} ‐colorable if it has a diagram admitting a non‐trivial \mathb {Z} ‐coloring.
In the following, we call the integers of the image of a \mathb {Z}‐coloring colors as usual.
The links illustrated in Figures 1 and 2 are examples those are \mathb {Z}‐colorable. Throughout

this paper, we adopt the names of links as those given in [2].
2a3a_{1}^ \backslh/^{!}\backslh_{\backslh}
Figure 1: L8n6(a\geq 1)

2

Figure 2: Pretzel link P (n, -n, \cdots , n, -n) for n\geq 1

There are a lot of researches on Fox colorings of links. Among them, in [4], Harary
and Kauffman originally defined the minimal coloring number for a link as the minimal
number of colors used in Fox colorings of the link. Since then, it has been studied in details
by many researchers. Actually, the minimal numbers of colors for n ‐colorable knots andlinks behave interestingly, and are often hard to determine.
Here we define the minimal coloring number for \mathb {Z}‐colorable links as a generalization.

Definition 2. Let us consider the cardinality of the image of a non‐trivial \mathb {Z} ‐coloring on
a diagram of a \mathb {Z} ‐colorable link L . We call the minimum of such cardinalities among allnon‐trivial \mathb {Z}‐colorings on all diagrams of L the minimal coloring number of L , and denote
it by mincol_{\mathbb{Z}}(L) .
In Section 2, we collect some preliminary results on \mathb {Z} ‐colorings. For example, we seethat the minimal coloring number of any splittable \mathb {Z} ‐colorable link is shown to be 2, andfor a non‐splittable \mathb {Z} ‐colorable link L , we show that mincol_{Z}(L)\geq 4 . That is, there areno links with the minimal coloring number 3.
Next, based on observations for \mathb {Z} ‐colorable link with at most 9 crossings, we introduce
a simple \mathb {Z} ‐coloring in Section 3, and show that if a link L admits a simple \mathb {Z}‐coloring,thenmincol_{Z}(L)=4.

In Section 4, we actually show that any non‐splittable \mathb {Z} ‐colorable link has a diagramwith a simple \mathb {Z} ‐coloring, and its minimal coloring number is always four. Remark that
this result is also proved by Meiqiao Zhang, Xian’an Jin and Qingying Deng almost

independently in [9].
In the proof of the above result, we give a procedure to obtain a diagram with a\mathb {Z} ‐coloring of four colors from any given diagram with a non‐trivial \mathb {Z} ‐coloring of a non‐splittable \mathb {Z} ‐colorable link. However, from a given diagram of a \mathb {Z}‐colorable link, by using

3
the procedure given in the our proof, the obtained diagram and \mathb {Z} ‐coloring might be very
complicated.
In Section 5, we give “simple” diagrams with \mathb {Z} ‐colorings of four colors for some par‐ticular class of \mathb {Z} ‐colorable link. In fact, we consider the link obtained by replacing eachcomponent of the given link with several parallel strands, which we call a parallel of alink.

Finally, in Section 6, we consider the question, for a Z ‐colorable link, how many colors
are necessary for their “simple” diagrams. Actually, we consider the minimal coloring
numbers of minimal diagrams of \mathb {Z} ‐colorable links, that is, the diagrams representing
the link with least number of crossings. We first show that, for any positive integer N,
there exists a non‐splittable \mathb {Z} ‐colorable link with a minimal diagram admitting only \mathb {Z} ‐colorings with at least N colors. In fact, the examples are given by families of pretzel
links; P (n, -n, n, -n, \cdots , n, -n) with at least 4 strands, P(-n, n+1, n(n+1)) with apositive integer n . On the other hand, by considering some particular subfamily, as acorollary, we have the following. There exists an infinite family of \mathb {Z} ‐colorable pretzel
links each of which has a minimal diagram admitting a \mathb {Z} ‐coloring with only four colors.
Also we give such examples given by some of \mathb {Z} ‐colorable torus links.
2

Preliminaries

In this section, we prepare some basic properties of \mathb {Z} ‐colorings.
Lemma 1. For any \mathb {Z} ‐colorable link, there exists a \mathb {Z} ‐coloring \gamma such that Im(\gamma)=\{0, a_{1}, a_{2}, \cdot\cdot\cdot , a_{n}\} with a_{i}>0 (i=1,2, \cdots , n) for some positive integer n.

Lemma 2. For a \mathb {Z} ‐coloring \gamma with 0= \min Im(\gamma), if an over arc at a crossing is coloredby 0 , then the under arcs at the crossing are colored by 0.
Lemma 3. For a \mathb {Z} ‐coloring \gamma with M= \max Im(\gamma) , if an over arc at a crossing iscolored by M , then the under arcs at the crossing are colored by M.
It is seen that any splittable link L is \mathb {Z} ‐colorable and mincol_{\mathbb{Z}}(L)=2 . On the other
hand, by using above lemmas, we see that the next holds for non‐splittable links.
Theorem 1. Let L be a non‐splittable \mathb {Z} ‐colorable link. Then mincol_{\mathbb{Z}}(L)\geq 4.

If a link is \mathb {Z} ‐colorable with four colors, we can show the following.
Theorem 2. If mincol_{Z}(L)=4 for a \mathb {Z} ‐colorable link L , then there exists a diagram Dof L and a \mathb {Z} ‐coloring \gamma on D such that Im(\gamma)=\{0,1,2,3\}.

3

Simple coloring

Among links of crossing numbers at most 9, there are only 5 links with zero determinant.

For such \mathb {Z} ‐colorable links, the colorings on the diagrams in [2] are quite distinctive.

In this section, we focus on the “simplest” \mathb {Z} ‐coloring found for the links with at most
9 crossings. Based on such examples, we introduce the following notion.

4
Definition 3. Let L be a non‐trivial \mathb {Z} ‐colorable link, and \gamma a\mathbb{Z} ‐coloring on a diagram Dof L . We call \gamma a simple \mathb {Z} ‐coloring if there exists an integer d such that, at each crossingin D , the difference between the colors of the over arc and the under arcs is d or 0.
For example, a pretzel knot P (n, -n, n, -n, \cdots , n, -n) with integer n admits a simple
See Figure 2.
For the links with simple \mathb {Z} ‐colorings, we have the following.

\mathb {Z} ‐coloring.

Theorem 3. [6, Theorem 4.2] Let L be a non‐splittable \mathb {Z} ‐colorable link. If there exists
a simple \mathb {Z} ‐coloring on a diagram of L , then mincol_{\mathbb{Z}}(L)=4.
The operation illustrated in Figure 3 is a key of the proof of the theorem.
hS

M‐d

A

hiA

M‐d

\underline{M-2d}|\underline{M}|\underline{M-}2d

2d
arrow

Md|

M-d|

Figure 3:

However, there are many diagrams of \mathb {Z}‐colorable links without simple \mathb {Z} ‐colorings. See
Figure 4 for example.

Figure 4: Non‐simple \mathb {Z} ‐coloring for L10n32

4

Minimal coloring number is four

The next is the main result in this article.

5
Theorem 4 ([8]). The minimal coloring number of any non‐splittable \mathb {Z} ‐colorable link isequal to 4.

This result is also proved by Meiqiao Zhang, Xian’an Jin and Qingying Deng almost

independently in [9]. Previously Zhang gave us her manuscript for her Master thesis. In
Zhang’s thesis, she calls a crossing an n ‐diff crossing if |b-a| and |b-c| are equal to n,

where the over arc is colored by b and the under arcs are colored by a and c by a \mathb {Z} ‐coloring\gamma at the crossing. Then she showed that if a \mathb {Z} ‐colorable link has a diagram with a 1‐diffcrossing, the link has a diagram with only 0 ‐diff crossings and 1‐diff crossings. Our proofof the theorem above is based on her arguments.
The proof of the theorem is achieved by giving a procedure to modify a diagram ofa non‐splittable \mathb {Z} ‐colorable link with a non‐simple \mathb {Z} ‐coloring to the one with a simple
coloring. One of the key moves is illustrated in Figure 5. We here omit the details.

t1
d_{m}

Figure 5:

By Theorem 4, any non‐splittable \mathb {Z} ‐colorable link has a diagram with a \mathb {Z}‐coloring of
4 colors. However, from a given diagram of a \mathb {Z} ‐colorable link, by using the procedure
given in the our proof of Theorem 4, the obtained diagram and \mathb {Z} ‐coloring might be very
complicated.
5

Parallel of link

In this section, we give a simple way to obtain a diagram which attains the minimalcoloring number for a particular family of \mathb {Z} ‐colorable links. That is, we consider thelink obtained by replacing each component of the given link with several parallel strands,
which we call a parallel of a link, as follows.

6
Definition 4. Let L=K_{1}\cup\cdots\cup K_{c} be a link with c components and D a diagram of L.
For a set (n_{1}, \cdots , n_{c}) of integers n_{i}\geq 1 , we denote by D^{(n_{1},\cdots,n_{c})} the diagram obtained bytaking n_{i}|‐parallel copies of the i‐th component K_{i} of D on the plane for 1\leq i\leq c . The
link L^{(n_{1},\cdots,n_{c})} represented by D^{(n_{1},\cdots,n_{c})} is called the (n_{1} , n_{c}) ‐parallel of L . When L
is a knot, that is c=1 , we call (n) ‐parallel L^{(n)} simply a n ‐parallel, and denote it by L^{n}.
We call a 2‐parallel of a knot untwisted if the linking number of the 2 components of theparallel is 0.
Examples of (n_{1} , n_{c}) ‐parallels of links are shown in Figures 6 and 7.

Figure 6: A(3,2) ‐parallel of the Hopf link

Figure 7: A 2‐parallel of the trefoil

We show that an even parallel of a link is \mathb {Z} ‐colorable except for the case of 2 parallels
with non‐zero linking number.

Theorem 5. (i) For a non‐trivial knot K and any diagram D of K that the writhe is 0,D^{2} always represents a \mathb {Z} ‐colorable link. Moreover, there exists a diagram D_{0} of K suchthat D_{0}^{2} is locally equivalent to a minimally \mathb {Z} ‐colorable diagram. (ii) Let L be a non‐splittable c ‐component link and D any diagram of L. For any even number n_{1} ,, n_{c}at least 4, D^{(n_{1},\cdots,n_{c})} always represents a \mathb {Z} ‐colorable link and is locally equivalent to aminimally \mathb {Z} ‐colorable diagram.
Here we give the definitions used in Theorem 5.
Definition 5. Let L be a \mathb {Z} ‐colorable link, and D a diagram of L. D is called a minimally\mathb {Z} ‐colorable diagram if there exists a \mathb {Z} ‐coloring \gamma on D such that the image of \gamma is equal
to the minimal coloring number of L.

7Definition 6. For diagrams D and D' of L,
mutually disjoint open subsets on \mathbb{R}^{2}U_{1}, U_{2} ,Reidemeister moves only in \bigcup_{i=1}^{m}U_{i}.

locally equivalent to D' if there exist, U_{n} such that D' is obtained from D by

D is

To prove Theorem 5 (i), we prepare the next lemma about the linking number ofcomponents of 2‐parallel of a knot.

Lemma 4. Let D a diagram of a knot K. For a 2‐parallel K^{2}=K_{1}\cup K_{2} represented byD^{2} , the linking number of K_{1} and K_{2} is equal to the writhe of D.
For non‐splittability of parallels of knots and links, we can also show the next.

Lemma 5. (i) Any n ‐parallel of a non‐trivial knot is non‐splittable. (ii) Any (n_{1} , n_{c}) ‐parallel of a non‐splittable link is non‐splittable.
To prove Theorem 5, we actually give \mathb {Z}‐colorings with four colors on the paralleleddiagrams. For the case of (ii), the colorings are illustrated in Figures 8, 9, and 10 withmodifications depicted in Figure 11.
()\underline{0}{\imath}^{:}\underline{1}0\underline{0}Figure 8:n_{i}\{\begin{ar ay}{l } 0- -0 0- \end{ar ay}
-0

1-1-

-0-1-10−-0

0−

Figure 9: n_{j}=4m for some integer m8n_{i}\{\begin{ar ay}{l } 0- -0 0-1-1- \end{ar ay}
-0

-1
-1

0−

-0

0−:

-0

Figure 10: n_{j}=4m+2\underlin{\underlin{-1}0|^{1}\underlin{2\underlin{3}|^1\underlin{0}Figure 11: Delete the color 3

6

Minimal diagrams

In this section, we consider the minimal coloring numbers of minimal diagrams of \mathb {Z} ‐colorable links, that is, the diagrams representing the link with least number of crossings.
6.1

Pretzel links

In this section, we first prove the next theorem.
Theorem 6. For an even integer n\geq 2 , the pretzel link P (n, -n, \cdots , n, -n) with at least
4 strands has a minimal diagram admitting only \mathb {Z} ‐colorings with n+2 colors.

Here a pretzel link P (a_{1} . , a_{n}) is defined as a link admitting a diagram consistingrational tangles corresponding to 1/a_{1},1/a_{2} ,, 1/a_{n} located in line. Such a pretzel link
P (n, -n, \cdots , n, -n) is known to be non‐splittable if n\geq 2 and the number of strands is
at least 4.

The pretzel link

P (n, -n, \cdots , n, -n)link. See [1] for example.

is \mathb {Z} ‐colorable since its determinant is 0 for the

We consider the diagram of P (n, -n, \cdots , n, -n) illustrated in Figure 12, which is aminimal diagram of the link due to the result in [7]. For this diagram, as shown in the
figure, we can find a \mathb {Z} ‐coloring with the colors 0, a, 2a ,, (n+1)a . It can be shown9that any \mathb {Z} ‐coloring must have such colors, that is, the minimal coloring number of the
diagram is equal to n+2.

(n‐

1) a

Figure 12:

On the other hand, by considering some particular subfamily, as a corollary, we seethat there exists an infinite family of \mathb {Z} ‐colorable pretzel links each of which has a minimal
diagram admitting a \mathb {Z} ‐coloring with only four colors as follows.
We consider the pretzel link P (2, -2,2, -2, \cdots , 2, -2) . The diagram depicted in Fig‐ure 13 is a minimal diagram by [7]. On the other hand, the \mathb {Z} ‐coloring given in the figure
has only four colors \{0,1,2,3\}.

Figure 13:

Next we consider the pretzel link P(-n, n+1, n(n+1)) for an integer n\geq 2 , and show
the following.
Theorem 7. For an integer n\geq 2 , the pretzel link P(-n, n+1, n(n+1)) has a minimal
diagram admitting only \mathb {Z} ‐colorings with n^{2}+n+3 colors.

Such pretzel links are all \mathb {Z} ‐colorable by [1] for example. In fact, the determinant of theP(-n, n+1, n(n+1)) is calculated as |(-n)\cdot(n+1)+(-n)\cdot n(n+1)+(n+1)\cdot n(n+1)|=

link
0.

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This theorem can be proved in the same way as for Theorem 6.

2n+

3n+

(n‐1) (n+1) n(n+2)-1

Figure 14:

6.2

Torus links

In this subsection, we consider torus links, that is, the links which can be isotoped onto
the standardly embedded torus in the 3‐space. By T(a, b) , we mean the torus link runninga times meridionally and b times longitudinally.

Theorem 8. For even integer n>2 and non‐zero integer p , the torus link T(pn, n) has
a minimal diagram admitting a \mathb {Z} ‐coloring with only four colors.

To prove the theorem above, we actually give \mathb {Z} ‐colorings with four colors on thestandard diagrams of torus links. We here only include a figure (Figure 15) to give suchcolorings.

..

...

1

Figure 15:]]11

Acknowledgement
We would like to thank Meiqiao Zhang for useful discussions. We also thank Akio
Kawauchi for giving me the motivation that I consider about parallels, and to Kouki
Taniyama for pointing out the necessity of Lemmas 5. We are also grateful to Ayumu
Inoue, Takuji Nakamura and Shin Satoh for useful discussions and advices. Finally wewould like to thank to the anonymous referee for his/her careful reading and pointing out

a gap in our proof of a theorem in [6].

References[1] O. T. Dasbach, D. Futer, E. Kalfagianni, X.‐S. Lin, and N. W. Stoltzfus, Alter‐
nating sum formulae for the determinant and other link invariants, J. Knot Theory

Ramifications 19 (2010), no. 6, 765‐782.

[2] J. C. Cha and C. Livingston, LinkInfo:
Table
http://www. indiana.edu/Nlinkinfo, (January 13, 2016).

of

Knot

Invariants,[3] R. H. Fox, A quick trip through knot theory, in Topology of 3‐manifolds and re‐lated topics (Proc. The Univ. of Georgia Institute, 1961), 120‐167, Prentice Hall,Englewood Cliffs, NJ.

[4] F. Harary and L. H. Kauffman, Knots and graphs. I. Arc graphs and colorings, Adv.
in Appl. Math. 22 (1999), no. 3, 312‐337.

[5] K. Ichihara and E. Matsudo, A lower bound on minimal number of colors for links,
Kobe J. Math. 33 (2016), no. 1‐2, 53‐60.
[6] K. Ichihara and E. Matsudo, Minimal coloring number for Z ‐colorable links, J. Knot
Theory Ramifications 26 (2017), no. 4, 1750018, 23 pp.

[7] W. B. R. Lickorish and M. B. Thistlethwaite, Some links with nontrivial polynomials
and their crossing‐numbers, Comment. Math. Helv. 63 (1988), no. 4, 527‐539.
[8] E.

Matsudo,Minimal coloring number for\mathb {Z} ‐colorable

links II, preprint,

arXiv: 1705. 07567v3[9] M. Zhang, X. Jin and Q. Deng, The minimal coloring number of any non‐splittable\mathb {Z} ‐colorable link is four, J. Knot Theory Ramifications 26 (2017), no. 13, 1750084, 18
pp.

Department of Mathematics
College of Humanities and Sciences
Nihon University
3‐25‐40 Sakurajosui, Setagaya‐ku, Tokyo 156‐8550
JAPAN

E‐‐mail address: ichihara@math. chs.nihon‐u.ac.jp

12
Graduate School of Integrated Basic Sciences,
Nihon University,
3‐25‐40 Sakurajosui, Setagaya‐ku, Tokyo 156‐8550
JAPAN

E‐mail address: cher16001@g.nihon‐u.ac.jp

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