An Improvement of the lower bound for the minimum number of link colorings by quandles H. Abchir

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An Improvement of the lower bound for the minimum

number of link colorings by quandles

H. Abchir

Hassan II University. Casablanca, Morocco.

e-mail: hamid.abchir@univh2c.ma

S. Lamsifer

Hassan II University. Casablanca, Morocco.

e-mail: soukaina.lamsifer-etu@etu.univh2c.ma

October 14, 2022

Abstract

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 kof the re-

duced 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.

1 Introduction

The idea of coloring knots was initiated by R. Fox around 1960 (see [7]). He introduced

colorings by dihedral quandles called n-colorings. Let Kbe a p-colorable knot, where p

is prime and let Cp(K)denote the minimal number of colors needed to color a diagram

of K. Nakamura et al. proved in [18] that Cp(K)≥ blog2pc+ 2.

The problem of ﬁnding the minimum number of colors for p-colorable knots with

primes up to 19 was investigated by many authors. In 2009, S. Satoh showed in [24]

that C5(K)=4. In 2010, K. Oshiro proved that C7(K)=4[21]. In 2016, T. Nakamura,

Y. Nakanishi and S. Satoh showed in [19] that C11(K) = 5. In 2017, M. Elhamdadi and

arXiv:2210.06530v1 [math.GT] 12 Oct 2022

J. Kerr [6] and independently F. Bento and P. Lopes [5] proved that C13(K) = 5. In 2020,

H. Abchir, M. Elhamdadi and S. Lamsifer [1] showed that C17(K) = 6. In 2022, Y. Han

and B. Zhou showed that C19(K)=6[9].

The same problem may be studied for colorings by linear Alexander quandles which

generalize dihedral ones. A linear Alexander quandle Λn,m is a quandle whose under-

lying set is n,n≥3, endowed with the binary operation x∗y=mx+(1−m)ymod n,

for some integer msuch that (m, n)=1. Let Lbe a link which admits a non-trivial color-

ing by Λn,m, i.e. for which there exists a non-constant quandle homomorphism from the

fundamental quandle Q(L)to Λn,m. We denote by mincoln,m(L)the minimum number

of colors needed to provide a non-trivial coloring of L. It is an interesting invariant of

L. For a knot K, if nis a prime integer p, L. Kauffman and P. Lopes showed in [12] that

2 + blogMpc ≤ mincolp,m(K),

where M=max{|m|,|m−1|}.

We give an enhancement of the last result by proving the following theorem.

Theorem 1.1. Let Kbe a knot. Let ∆0

K(t) =

i=0

citibe the reduced Alexander polynomial

of K. Let mbe an integer, such that m > max

0≤i≤k{|ci|} + 1 and p= ∆0

K(m)is an odd prime

integer.

1. If ck= 1 and the penultimate non-zero coefﬁcient is negative, then

k+ 1 ≤mincolp,m(K).

2. If ck>1or the penultimate non-zero coefﬁcient is positive, then

k+ 2 ≤mincolp,m(K).

So, the lower bound stabilizes for suitable choices of mand pand no longer depends

on these two integers. On the other hand, the lower bound we give comes from a

topological invariant of the knot. In particular, it is exactly k+ 1 for an L-space knot, for

any m > 1. This entails that if the torus knot T(a, b)whose crossing number is c(T(a, b))

admits a non-trivial coloring by Λp,m, then mincolp,m(T(a, b)) is bounded as follows:

c(T(a, b)) −(a−2) ≤mincolp,m(T(a, b)) ≤c(T(a, b)).

Hence mincolp,m(T(2, b)) = c(T(2, b)). On the other hand, we show that for suitable

choices of mand p,T(2, b)displays KH behavior, which means that T(2, b)admits a

reduced alternating diagram equipped with a non-trivial coloring by the quandle Λp,m

such that different arcs receive different colors. Furthermore, we show that if the Pret-

zel knot P(−2,3, a)has a non-trivial coloring by a linear Alexander quandle Λp,m, then

for suitable choices of mand pwe have a+ 4 ≤mincolp,m(P(−2,3, a)) and the equality

holds for m= 2. Finally, we show that Theorem 1.2 proved by Kauffman and Lopes in

[12] holds also for links with more than one component.

The paper is organized as follows. In the second section, we recall the main tools we

need to prove our results. In the third section we prove our main theorem. The fourth

section is devoted to some applications. In the last section, we give a generalization of

Theorem 1.2 in [12] to links.

2 Preliminaries

In this section we give an overview of the main tools we need.

Quandles.

Deﬁnition 2.1. A quandle, is a non-empty set Qequipped with a binary operation

∗:Q×Q−→ Q

(x, y)7−→ x∗y

satisfying the following three axioms:

1. For all x∈Q,x∗x=x.

2. For all y∈Q, the map Ry:Q→Qdeﬁned by Ry(x) = x∗y,x∈Q, is bijective.

3. For all x, y, z ∈Q,(x∗y)∗z= (x∗z)∗(y∗z)(right self-distributivity).

We write x∗−1yfor R−1

y(x).

When needed, we will denote the quandle Qby the pair (Q, ∗).

Example 2.1. Trivial quandle.

Let Xbe a non-empty set with the operation x∗y=xfor any x, y ∈X(i.e. Ry=idX, for any

y∈X), is a quandle called the trivial quandle.

Example 2.2. Dihedral quandle.

Let nbe a positive integer. For xand yin n(integers modulo n), deﬁne x∗y= 2y−x

mod n. The operation ∗deﬁnes a quandle structure on n, called the dihedral quandle and is

sometimes denoted Rn.

Example 2.3. Alexander quandle.

An Alexander quandle Mis a [t, t−1]-module endowed with the following binary operation:

x∗y=tx + (1 −t)yfor all x, y ∈M.

Note that we have x∗−1y=t−1x+ (1 −t−1)y.

Example 2.4. Linear Alexander quandle.

Let nbe an integer n > 1, one can consider Λn/(h)where Λn=n[t, t−1]and his a monic

polynomial in t(see [20]). If h(t)=(t−m)and mand nare integers such that gcd (m, n) = 1,

then we obtain what is called linear Alexander quandle that we denote Λn,m. This amounts to

considering the underlying set nwith the binary operation:

x∗y=mx + (1 −m)y(mod n),for all x, y ∈Zn.

Note that we have x∗−1y=m−1x+ (1 −m−1)y(mod n).

If m=−1, we get the dihedral quandle Rn.

Deﬁnition 2.2. Let (Q1,∗1)and (Q2,∗2)be two quandles. A map f:Q1→Q2is a quandle

homomorphism if it satisﬁes

f(x∗1y) = f(x)∗2f(y),∀x, y ∈Q1.

If fis bijective, we say that fis a quandle isomorphism. If fis bijective and Q1=Q2, the map

fis called a quandle automorphism.

Coloring links by Alexander quandles

Let (Q, ∗)be a quandle and Da diagram of an oriented link L. A coloring of Dby Q

is a map Cfrom the set of arcs of Ddenoted by Ato Q, such that at each crossing of

D, if the over-arc α1is colored by C(α1) = yand the incoming under-arc is colored by

C(α2) = xthen the outcoming under-arc is colored by C(α3) = x∗yor C(α3) = x∗−1y

according to the rule depicted in Fig.1.

Figure 1: Coloring conditions.

If Qis an Alexander quandle, by collecting the coloring conditions at all crossings

of D, we get a homogeneous system of linear equations over [t, t−1]. The matrix asso-

ciated to this system of equations is called the Alexander matrix. Its rows correspond to

the crossings of Dand the columns correspond to the arcs of D. Each row has only three

non-zero entries which are t,1−tand −1. So on the one hand (λ, . . . , λ)is a solution

for any λ∈[t, t−1](trivial solutions), and on the other hand the determinant of the

Alexander matrix is zero. Hence, it is easy to see that a non-trivial solution of the initial

homogeneuous system corresponds to a non-trivial solution of the system of equations

determined by the original matrix with one row and one column deleted. The determi-

nant of this last submatrix is known to be the Alexander polynomial of the considered

link denoted by ∆L(t). Therefore, the existence of non-trivial solutions corresponds to

working on the quotient of [t, t−1]by ∆L(t), which is a Laurent polynomial on the

variable tdetermined up to ±tn, for any integer n. We will use the reduced Alexander

polynomial deﬁned in [4].

Remarks 2.1. 1. If Lis a knot K, then the reduced Alexander polynomial is exactly that

given in the proof of Corollary 6.11 in [14], ∆L(t) = c0+c1t+c2t2+· · · +cNtN, where

Nis even, cN−r=cr,cN

2is odd and c0>0.

2. If Lhas µcomponents, µ≥2, the reduced Alexader polynomial is obtained from the mul-

tivariable Alexander polynomial ∆L(t1, . . . , tµ)by setting ti=tfor each i. Recall that

there is a relation between Alexander polynomial and multivariable Alexander polynomial

as shown in Proposition 7.3.10 in [13]:

∆L(t) = (1 −t)∆L(t, . . . , t).

Example 2.5. We consider the diagram Dof the knot 73whose arcs are labeled as shown in

Fig.2. By writing the coloring conditions illustrated in Fig.1 at each crossing ciof D,1≤i≤7,

we obtain the homogeneuous system of linear equations shown on the right side of Fig.2,

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AnImprovementofthelowerboundfortheminimumnumberoflinkcoloringsbyquandlesH.AbchirHassanIIUniversity.Casablanca,Morocco.e-mail:hamid.abchir@univh2c.maS.LamsiferHassanIIUniversity.Casablanca,Morocco.e-mail:soukaina.lamsifer-etu@etu.univh2c.maOctober14,2022AbstractWeimprovethelowerboundfortheminimumnumb...

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An Improvement of the lower bound for the minimum number of link colorings by quandles H. Abchir

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