Multivariate Alexander quandles V. Constructing the medial quandle of a link Lorenzo Traldi

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Multivariate Alexander quandles, V.

Constructing the medial quandle of a link

Lorenzo Traldi

Lafayette College

Easton, PA 18042, USA

traldil@lafayette.edu

Abstract

We explain how the medial quandle of a classical or virtual link can

be built from the peripheral structure of the reduced Alexander module.

Keywords: Alexander module; longitude; medial quandle.

Mathematics Subject Classiﬁcation 2020: 57K10

1 Introduction

This is the ﬁfth paper in a series involving Alexander modules and quandles,

but the paper is written so that familiarity with the earlier papers in the series is

not required. The purpose of the paper is to show that the medial quandle of a

link can be constructed from the reduced Alexander module. The construction

requires the module’s peripheral elements, which were introduced recently [10].

In the case of a knot, the peripheral elements do not contribute anything to

the construction, and we deduce an extension to virtual knots of Joyce’s result

[5, Theorem 17.3] that the medial quandle of a classical knot is the standard

Alexander quandle on the Alexander invariant of the knot.

To keep the introduction as short as possible, we leave the statements of

many deﬁnitions and previous results for later in the paper. We should say,

though, that we use the terms knots and links for oriented virtual knots and

links.

Let Λ = Z[t±1] be the ring of Laurent polynomials in the variable t, with

integer coeﬃcients. Here is the central algebraic construction of the paper.

Deﬁnition 1. Suppose Mis a Λ-module with a submodule N,Iis a nonempty

set, mi∈M∀i∈I, and mi−mj∈N∀i, j ∈I. Suppose also that for each

i∈I,Xiis a submodule of Nsuch that (1 −t)·Xi= 0. Then there is an

associated medial quandle Q(N, (mi),(Xi)), deﬁned as follows.

1. For each i∈I, let Qi=N/Xi. If i̸=j∈Ithen the sets Qiand Qjare

arXiv:2210.04611v5 [math.GT] 3 Sep 2024

understood to be disjoint. Deﬁne the set Q(N, (mi),(Xi)) by

Q(N, (mi),(Xi)) = [

i∈I

Qi.

2. Deﬁne an operation ▷on Q(N, (mi),(Xi)) as follows. If i, j ∈I,x∈Qi

and y∈Qj, then

x▷y=mj−mi+tx + (1 −t)y+Xi∈Qi.

Here are three comments on Deﬁnition 1. First: the formula for ▷in part 2

of Deﬁnition 1 is well deﬁned, because the fact that (1 −t)·Xj= 0 guarantees

there is no ambiguity in the value of the coset (1 −t)y+Xi. Second: Deﬁnition

1 allows Mto strictly contain N, but this is a merely stylistic choice. If we

would like to have M=N, we can always replace the elements mi∈Mwith

elements m′

i∈N, by choosing a ﬁxed index i0∈Iand deﬁning m′

i=mi−mi0

∀i∈I. It is easy to see that then Q(N, (mi),(Xi)) ∼

=Q(N, (m′

i),(Xi)). Third:

If µis a positive integer and I={1, . . . , µ}, we use the notation Q(N, (mi),

(Xi)) = Q(N, m1, . . . , mµ, X1, . . . , Xµ).

In Sec. 3 we show that Deﬁnition 1 provides a general description of medial

quandles. To be precise:

Theorem 2. If Qis a medial quandle, then there are M, N, I, (mi)and (Xi)

such that Q∼

=Q(N, (mi),(Xi)).

We should mention that Deﬁnition 1 and Theorem 2 are closely related to the

structure theory of medial quandles due to Jedliˇcka et al. [3, 4]. In particular,

Theorem 2 extends and reﬁnes [4, Lemma 4.3].

The reduced Alexander module Mred

A(L) is a well-known invariant of a link

L. For a classical link, it corresponds to the ﬁrst relative homology group of the

total linking number cover, with respect to its ﬁber. The module is conveniently

described by a presentation with generators and relations corresponding to the

arcs and crossings of a link diagram (respectively). There is a Λ-linear map

ϕred

L:Mred

A(L)→Λ, under which 1 is the image of every module generator

corresponding to an arc in a diagram. The kernel of ϕred

Lis the reduced Alexander

invariant of L.

In recent work [10], we introduced peripheral elements in the reduced Alexan-

der module of a link. If L=K1∪ · · · ∪ Kµis a µ-component link then there are

subsets M1(L), . . . , Mµ(L)⊂Mred

A(L); the elements of Mi(L) are the meridians

of Kiin Mred

A(L). In a module presentation of Mred

A(L) corresponding to a dia-

gram of L, every generator corresponding to an arc of Kiis a meridian associated

with Ki; also, every meridian is mapped to 1 by ϕred

L. For each i∈ {1, . . . , µ}

there is a single longitude χi(L) corresponding to Ki. These longitudes have

several distinctive properties, including χi(L)∈ker ϕred

Land (1 −t)·χi(L) = 0.

The list

(Mred

A(L), M1(L), . . . , Mµ(L), χ1(L), . . . , χµ(L))

is the enhanced reduced Alexander module of L; it is denoted Menr

A(L). The

enhanced reduced Alexander module is a link invariant, in the sense that if L

and L′are equivalent links then there is a Λ-linear isomorphism Mred

A(L)→

Mred

A(L′) such that for each i∈ {1, . . . , µ}, the image of Mi(L) is Mi(L′) and

the image of χi(L) is χi(L′).

Here is the main theorem of this paper.

Theorem 3. Let Lbe a link. For each i∈ {1, . . . , µ}, let mi∈Mi(L)be

any meridian of Kiin Mred

A(L), and let Xibe the cyclic submodule of Mred

A(L)

generated by χi(L). Then the quandle Q(ker ϕred

L, m1, . . . , mµ, X1, . . . , Xµ)of

Deﬁnition 1 is isomorphic to the medial quandle of L.

We should mention that the medial quandle of Lwas denoted AbQ(L) in

Joyce’s seminal work [5]. We use the notation MQ(L) instead.

In the special case when Lis a knot (i.e. µ= 1), the longitude χ1(L) is

0 [10]. Theorem 3 then states that MQ(L) is the quandle on ker ϕred

Lgiven

by the operation x ▷ y =tx + (1 −t)y. This is the standard way to deﬁne a

quandle structure on a Λ-module; in the literature such a quandle is called an

aﬃne quandle or an Alexander quandle. We deduce the following extension of

a classical result of Joyce [5, Theorem 17.3] to virtual knots.

Corollary 4. Suppose Lis a knot. Then the medial quandle of Lis isomorphic

to the standard Alexander quandle on the Λ-module ker ϕred

L. Moreover, MQ(L)

and ker ϕred

Lare equivalent as invariants of L, i.e. each invariant determines the

other.

In general, Theorem 3 implies that the medial quandle of a link is always

determined by the enhanced reduced Alexander module Menr

A(L). When µ > 1,

Menr

A(L) is a strictly stronger link invariant than MQ(L). There are two sim-

ple reasons for this. The ﬁrst reason is that Menr

A(L) involves the indexing

of the components of L=K1∪ · · · ∪ Kµ, and the medial quandle does not.

Therefore if L′is obtained from Lby permuting the indices of K1, . . . , Kµthen

Menr

A(L) might distinguish Lfrom L′, but the medial quandle cannot. The sec-

ond reason is that Menr

A(L) involves the individual longitudes χ1(L), . . . , χµ(L),

while Q(ker ϕred

L, m1, . . . , mµ, X1, . . . , Xµ) involves only the cyclic submodules

X1, . . . , Xµgenerated by χ1(L), . . . , χµ(L). Therefore if Menr

A(L) and Menr

A(L′)

diﬀer only because their longitudes are negatives of each other, then Menr

A(L)

might distinguish Lfrom L′, but the medial quandle cannot.

Here is an outline of the paper. In Sec. 2, we brieﬂy summarize some aspects

of the general theory of medial quandles. In Sec. 3, we discuss the special

properties of the quandles given by Deﬁnition 1, and prove Theorem 2. Notation

for link diagrams and link invariants is established in Sec. 4, and Theorem 3 is

proven in Sec. 5. In Sec. 6 we observe that the quandle Qred

A(L) of [11] is the

maximal semiregular image of MQ(L). In Sec. 7 we apply our discussion to the

involutory medial quandle IMQ(L), sharpening some results from [9]. Several

examples are presented in Sec. 8.

Before proceeding we should thank an anonymous reader for helpful com-

ments and corrections.

2 Medial Quandles

In this section we present some properties of medial quandles, without much

explanation or proof. For a more detailed and thorough account of the theory

we refer to Jedliˇcka et al. [3, 4]. Much of what we summarize here is also

detailed in the preceding paper in this series [11]. Before jumping in, we should

mention that in his seminal work [5], Joyce used the term “abelian” rather than

“medial.”

Deﬁnition 5. Amedial quandle is given by a binary operation ▷on a set Q.

The following properties must hold.

1. x▷x=x∀x∈Q.

2. For each y∈Q, a permutation βyof Qis deﬁned by βy(x) = x▷y.

3. (w ▷ x)▷(y ▷ z)=(w ▷ y)▷(x▷z)∀w, x, y, z ∈Q.

All the quandles we consider in this paper are medial, but it is worth men-

tioning that a general quandle is required to satisfy the ﬁrst two properties of

Deﬁnition 5, along with the special case of the third property in which y=z.

If Qis a medial quandle then the maps βy:Q→Qare the translations or

inner automorphisms of Q. We often use the notation β−1

y(x) = x ▷−1y. The

translations are automorphisms of Q, and the subgroup of Aut(Q) they gener-

ate is denoted β(Q). The composition βyβ−1

zof a translation with the inverse

of a translation is an elementary displacement of Q, and the subgroup of β(Q)

generated by the elementary displacements is the displacement group Dis(Q).

Note that the inverse of an elementary displacement is also an elementary dis-

placement, so every displacement is a composition of elementary displacements.

In fact it turns out that Dis(Q) includes all products Qβmi

yiwith Pmi= 0. As

discussed in [11, Sec. 5], the displacement group is a normal abelian subgroup

of β(Q), and it may be given the structure of a Λ-module by choosing any ﬁxed

element q∗∈Q, and deﬁning t·d=βq∗dβ−1

q∗∀d∈Dis(Q). Changing the choice

of q∗does not change the Λ-module structure of Dis(Q). The reason is simple:

if q∗, q∗∗ ∈Qthen βq∗β−1

q∗∗ ∈Dis(Q), so commutativity of Dis(Q) implies the

following.

βq∗dβ−1

q∗=βq∗dβ−1

q∗(βq∗β−1

q∗∗ )(βq∗β−1

q∗∗ )−1

= (βq∗β−1

q∗∗ )−1βq∗dβ−1

q∗(βq∗β−1

q∗∗ )

=βq∗∗ β−1

q∗βq∗dβ−1

q∗βq∗β−1

q∗∗ =βq∗∗ dβ−1

q∗∗

Here is an easy consequence of the fact that t·d=βq∗dβ−1

q∗does not depend

on the choice of q∗.

Lemma 6. If d∈Dis(Q)has a ﬁxed point, then d=t·d.

Proof. If d(q∗) = q∗then for every x∈Q, we have

dβq∗(x) = d(x▷q∗) = d(x)▷ d(q∗) = d(x)▷ q∗=βq∗d(x).

Therefore dβq∗=βq∗d, so d=βq∗dβ−1

q∗.

If Q1and Q2are quandles then a function f:Q1→Q2is a quandle map or

quandle homomorphism if the equality f(x ▷ y) = f(x)▷ f(y) is always satisﬁed.

(That is, fβy=βf(y)f∀y∈Q.) This equality implies

f(x ▷−1y)▷ f(y) = f((x ▷−1y)▷ y) = f(x),

so the equality f(x ▷−1y) = f(x)▷−1f(y) is also always satisﬁed. (That

is, fβ−1

y=β−1

f(y)f∀y∈Q.) The displacement groups of medial quandles

are functorial with respect to surjective homomorphisms, in the sense that

a surjective quandle map f:Q1→Q2induces a surjective homomorphism

Dis(f) : Dis(Q1)→Dis(Q2) of abelian groups, deﬁned in the natural way: If

y1, . . . , yn∈Q1,m1, . . . , mn∈ {±1}and Pmi= 0, then

Dis(f)Yβmi

yi=Yβmi

f(yi).

Note that if Qβmi

yi=d∈Dis(Q1) then as fis a quandle map,

Dis(f)(d)◦f=βm1

f(y1)· · · βmn−1

f(yn−1)βmn

f(yn)f

=βm1

f(y1)· · · βmn−1

f(yn−1)fβmn

=· · · =f βm1

y1· · · βmn−1

yn−1βmn

yn=f◦d.

If q∈Qthen the orbit of qin Qis the smallest subset of Qthat contains q

and is closed under the action of β(Q). We denote the orbit Qq. It turns out

that Qq={d(q)|d∈Dis(Q)}. In fact, if Fix(q) = {d∈Dis(Q)|d(q) = q}

then Fix(q) is a Λ-submodule of Dis(Q), and Qqis isomorphic to the stan-

dard Alexander quandle on the quotient module Dis(Q)/Fix(q). That is, the

quandle operation is given by x▷y =tx + (1 −t)y. An isomorphism can

be deﬁned in the natural way: if d∈Dis(Q) then d(q)∈Qqcorresponds to

d+ Fix(q)∈Dis(Q)/Fix(q). In particular, a medial quandle Qwith only one

orbit is isomorphic to the standard Alexander quandle on the Λ-module Dis(Q).

As a consequence of these properties, we have the following characterization

of isomorphisms of medial quandles.

Proposition 7. Let Qand Q′be medial quandles. Then a quandle map f:

Q→Q′is an isomorphism if and only if it has these three properties.

1. The map fis surjective.

2. Whenever x, y ∈Qhave diﬀerent orbits, f(x), f(y)∈Q′also have diﬀer-

ent orbits.

3. Whenever x∈Q,d∈Dis(Q)and Dis(f)(d)(f(x)) = f(x), it is also true

that d(x) = x.

Proof. If fis an isomorphism then the three listed properties certainly hold.

For the converse, we need to prove that the three properties imply that fis

injective.

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摘要：

MultivariateAlexanderquandles,V.ConstructingthemedialquandleofalinkLorenzoTraldiLafayetteCollegeEaston,PA18042,USAtraldil@lafayette.eduAbstractWeexplainhowthemedialquandleofaclassicalorvirtuallinkcanbebuiltfromtheperipheralstructureofthereducedAlexandermodule.Keywords:Alexandermodule;longitude;media...

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Multivariate Alexander quandles V. Constructing the medial quandle of a link Lorenzo Traldi

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