PRIMENESS OF ALTERNATING VIRTUAL LINKS THOMAS KINDRED Abstract. Using a new tool called lassos we establish a new

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PRIMENESS OF ALTERNATING VIRTUAL LINKS

THOMAS KINDRED

Abstract. Using a new tool called lassos, we establish a new

correspondence between cellular link diagrams on closed sur-

faces and equivalence classes of virtual link diagrams. This is

analogous to a well-known correspondence among the links rep-

resented by these diagrams, but with a crucial subtlety. We ex-

plain how, under these correspondences, the traditional notion

of primeness for virtual links is stricter than the one for links in

thickened surfaces. We extend a classical result of Menasco by

proving that an alternating link in a thickened surface is prime

in the stricter sense unless it is “obviously” composite. (Adams

et al and Howie–Purcell previously extended Menasco’s result

for the other notion of primeness.) We describe, given an al-

ternating virtual link diagram, how to determine by inspection

whether the virtual link it represents is prime in either sense.

1. Introduction

Traditionally, a nonclassical virtual link Kis said to be prime if

it cannot be decomposed as a nontrivial connect sum. On the other

hand, a nonstabilized1link Lin a thickened surface Σ ×Iof positive

genus is traditionally called prime if, for any pairwise connect sum

decomposition (Σ ×I, L) = (Σ ×I, L1)#(S3, L2), L2is trivial.

Interestingly, under a well-known correspondence between virtual

links and nonstabilized links in thickened surfaces, prime links in

one setting do not always correspond to prime links in the other.

For example, in Figure 1, the knot Kin a thickened surface of genus

2 is prime, but the corresponding virtual knot Lis not. We show:

Theorem 5.9. Given a nonsplit virtual link Kand the corresponding

nonstabilized link Lin a thickened surface Σ×I:

(1) (Σ, L)is prime (in the traditional sense, which we call “lo-

cal”) if and only if Kadmits no nontrivial connect sum de-

composition K=K1#K2in which K1is a classical link and

g(K2) = g(K); and

(2) Kis prime if and only if (Σ, L)is what we call “pairwise

prime” (see Deﬁnition 5.6).

1See §2.2 for deﬁnitions of terms including nonstabilized and cellular.

arXiv:2210.03225v2 [math.GT] 29 Aug 2024

2 THOMAS KINDRED

K:L:

Figure 1. A knot that is locally, but not pairwise, prime.

Figure 2. Converting the neighborhood of a virtual

link diagram to an abstract link diagram

The pairwise prime condition in part (2) is how Matveev deﬁnes

prime virtual links [Ma12], but he does not mention (or prove) that

this condition coincides with the natural deﬁnition in this paper’s

ﬁrst sentence; part (2) of Theorem 5.9 conﬁrms the equivalence of

these deﬁnitions.

Given a reduced cellular alternating diagram D⊂Σ of a link

L⊂Σ×I, one can determine by inspecting Dwhether or not Lis

prime in either sense. For local primeness, this was already known,

by work of Menasco and Adams et al [Me84, Aetal19], and Howie–

Purcell proved a stronger theorem about local primeness in [HP20].

We prove this for pairwise primeness:

Theorem 6.4. Given a reduced cellular alternating diagram D⊂Σ

of a link L⊂Σ×I,(Σ, L)is pairwise prime if and only if whenever

(Σ, D) = (Σ1, D1)#(Σ2, D2), either (Σ1, D1)or (Σ2, D2)is (S2,⃝).

To translate Theorem 6.4 to a statement about virtual links and

their diagrams, we use a well-known correspondence between virtual

links and stable equivalence classes of links in thickened surfaces

[Ka98, KK00, CKS02], together with a new correspondence between

the associated diagrams:

Correspondence 4.3. The following gives a bijection from equiva-

lence classes [V]of nonsplit virtual link diagrams under non-classical

R-moves to cellular link diagrams on connected closed surfaces:

Choose V∈[V], take a regular neighborhood νV of Vin S2, modify

νV near each virtual crossing of Vas shown in Figure 2,2and cap

oﬀ each boundary component (abstractly) with a disk.

At ﬁrst glance, this follows exactly the sort of construction de-

scribed in [Ka98, KK00, CKS02], but the reverse direction of the

2At this intermediate stage, we have an abstract link diagram, which we will

not need again.

PRIME VIRTUAL LINKS 3

correspondence contains a hidden subtlety, one which is fundamental

to understanding both correspondences and which, to the author’s

knowledge, has not previously been observed in the literature. We

describe this subtlety in §4.2.

Theorem 6.4 and the related results of Adams et al [Aetal19],

Howie–Purcell [HP20], and Menasco [Me84] state that an alternat-

ing link (in S3or a thickened surface) is composite (in an appropriate

sense) if and only if it is “obviously so” in a given reduced alternat-

ing diagram. Theorem 6.4 can be stated in the same manner. When

translating Theorem 6.4 and the related results of Adams et al and

Howie–Purcell to virtual link diagrams, however, “obvious” feels in-

accurate. What does it mean for an alternating virtual link diagram

Vto be “obviously” composite (in either the local or pairwise sense)?

Certainly, if Vdecomposes as a diagrammatic connect sum of two

nontrivial links, then it is obviously composite; but this is too re-

strictive (the theorem is untrue with such a strong requirement).

Our remedy is (to consider not just Vbut [V] and) use a new tool

called lassos to capture the salient features of [V].

Given a link diagram Don a closed surface Σ, a lasso is a disk

X⊂Σ that contains all crossings of D. Similarly, given a virtual link

diagram V⊂S2, a lasso is a disk X⊂S2that contains all classical

crossings of Vand no virtual ones. In both contexts, a lasso Xis

acceptable if the part of the diagram in Xis connected and the part

of the diagram outside Xdoes not admit an “obvious” simpliﬁca-

tion. In §§3.1-3.2, we deﬁne these terms carefully and establish basic

properties, like the fact that D⊂Σ admits has an acceptable lasso

if and only if Dis connected. In §3.3, we introduce lasso diagrams

and lasso numbers. A lasso diagram is essentially a virtual link di-

agram in which the virtual crossings are captured combinatorially

rather than shown (see Figure 4). The lasso number is an invariant

of virtual links. We establish some basic properties. Computing this

invariant seems like a challenging, but approachable, problem.

In §4, we use lassos to establish Correspondence 4.3, which we

then extend to equivalence classes of lasso diagrams under two types

of moves (see Moves 1-2 and Figure 8). This gives the following

extension of the well-known Correspondence 4.4:

Correspondence 4.9. There is a triple bijective correspondence be-

tween (i) virtual links, (ii) stable equivalence classes of links in thick-

ened surfaces, and (iii) equivalence classes of lasso diagrams under

Moves 1-2 and classical R-moves.

We also prove the following diagrammatic extension of Kuper-

berg’s theorem:

Theorem 4.7. All minimal genus diagrams of a nonsplit virtual link

are related by minimal-genus-preserving generalized R-moves.

4 THOMAS KINDRED

Section 6 addresses local and pairwise primeness for links in thick-

ened surfaces, culminating with Theorem 6.4, and §7 adapts that the-

orem to virtual link diagrams and lasso diagrams. Here, we ﬁnd that

nugatory crossings are always obvious in acceptable lasso diagrams,

even though nugatory crossings can be harder to identify in virtual

link diagrams. This is particularly important because, in the virtual

setting there are two types of nugatory crossings (removable and non-

removable), and such crossings provide the main technical obstacle to

translating statements about diagrammatic primeness (local or pair-

wise) to statements about prime virtual links. Everything connects

in our ﬁnal result:

Theorem 7.8. Let V′be a connected alternating diagram of a virtual

link K, and consider a diagram V∈[V′]which admits an acceptable

lasso X. Assume that Vhas at least one virtual crossing. Then:

(1) When V′has no nugatory crossings, Kis prime if and only

if V′is prime.

(2) When V′has no removably nugatory crossings, Kis locally

prime if and only if V′is locally prime.

Section 8 oﬀers some concluding thoughts.

2. Background

2.1. Notation. Throughout:

•We work in the piecewise-linear category.

•Σ denotes a closed orientable surface, not necessarily con-

nected or of positive genus.

•Iand I+respectively denote the intervals [−1,1] and [0,1].

•In Σ×I, we identify Σ with Σ×{0}and write Σ×{±1}= Σ±.

•πΣdenotes projection πΣ: Σ ×I→Σ.

•For a pair (Σ, L), Lis a link in Σ ×Iwhich intersects each

component of Σ ×I.

•For a pair (Σ, D), Dis a link diagram on Σ which intersects

each component of Σ.

•c

S3denotes S3\(2 points), which is identiﬁed homeomorphi-

cally with S2×R, and π:c

S3→S2denotes projection.

• |X|denotes the number of connected components of X.

•Given transverse submanifolds S, T of some ambient mani-

fold, the notations |S∩T|and |S⋔T|carry the same mean-

ing; we use the latter notation if we wish to emphasize or

clarify that Sand Tare transverse.

PRIME VIRTUAL LINKS 5

•In a manifold X, given a subset Ythat has a closed regular

neighborhood, we denote this neighborhood νY and its inte-

rior ◦

νY . Further, X\\Ydenotes “Xcut along Y,” which is

the metric closure of X\Y.3

2.2. Links in thickened surfaces. A pair (Σ, L) is stabilized if,

for some circle4γ⊂Σ, Lcan be isotoped so that it is disjoint from

the annulus γ×Ibut intersects each component of (Σ ×I)\(γ×I);

one can then destabilize the pair (Σ, L) by cutting Σ ×Ialong γ×I

and attaching two 3-dimensional 2-handles in the natural way (this

may disconnect Σ); the reverse operation is called stabilization. Note

conversely that (Σ, L) is nonstabilized if and only if every diagram D

of Lon Σ is cellular, meaning that Dcuts Σ into disks.

Convention 2.1. We regard two pairs (Σ, L) and (Σ′, L′) as equiva-

lent if there is a pairwise homeomorphism h: (Σ×I, L)→(Σ′×I, L′)

under which Σ+→Σ′

+, respecting orientations. We regard two pairs

(Σ, D) and (Σ′, D′) as equivalent if there is a pairwise homeomor-

phism (Σ, D)→(Σ′, D′) in which Σ →Σ′respects orientations and

D→D′respects crossing information.

Theorem 2.2 (Theorem 1 of [Ku03]).The stable equivalence class

of any pair (Σ, L)contains a unique nonstabilized representative.

We call a pair (Σ, L)split if Lhas a disconnected diagram on Σ. In

particular, (Σ, L) is split whenever Σ is disconnected. Equivalently,

(Σ, L) is split if, for some (possibly empty) disjoint union of circles

γ⊂Σ, Σ \γis disconnected, and Lcan be isotoped so that it is

disjoint from γ×Ibut intersects each component of (Σ×I)\(γ×I).

Kuperberg’s theorem implies that when (Σ, L) is nonsplit, (Σ, L)

is nonstabilized if and only if Σ has minimal genus in its stable equiv-

alence class. Note that when Σ is connected, if (Σ, L) is split, then

it is also stabilized. The converse is false. In fact, by Kuperberg’s

theorem, the number of split components is an invariant of stable

equivalence classes.

If Lis nonsplit and g(Σ) >0, then (Σ ×I)\Lis irreducible,

as Σ ×Iis always irreducible, since its universal cover is R2×R

[CSW14]. The converse of this, too, is false. Indeed, if (Σi×I, Li) is

nonsplit and g(Σi)>0 for i= 1,2, then construct (Σ1#Σ2, L1⊔L2)

3X\\Yis homeomorphic to X\◦

νY but may have extra structure from Y

encoded in its boundary. For example, if Fis a compact orientable surface in S3,

then S3\\Fis a sutured manifold: the extra structure here is the copy of ∂F

on ∂(S3\\F), which cuts ∂(S3\\F) into two copies of F. Similarly, if (Σ, D) is

cellular, then the boundary of each disk of Σ\\Dcontains a copy of each incident

edge and vertex (i.e. crossing) from D.

4We use “circle” as shorthand for “smooth simple closed curve.”

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PRIMENESSOFALTERNATINGVIRTUALLINKSTHOMASKINDREDAbstract.Usinganewtoolcalledlassos,weestablishanewcorrespondencebetweencellularlinkdiagramsonclosedsur-facesandequivalenceclassesofvirtuallinkdiagrams.Thisisanalogoustoawell-knowncorrespondenceamongthelinksrep-resentedbythesediagrams,butwithacrucialsubt...

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