THE VIRTUAL FLYPING THEOREM THOMAS KINDRED Abstract. We extend the flyping theorem to alternating links

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THE VIRTUAL FLYPING THEOREM

THOMAS KINDRED

Abstract. We extend the ﬂyping theorem to alternating links

in thickened surfaces and alternating virtual links. The proof

of the former result uses work of Boden–Karimi to adapt the

author’s geometric proof of Tait’s 1898 ﬂyping conjecture (ﬁrst

proved in 1993 by Menasco–Thistlethwaite), while the proof

of the latter involves a diagrammatic correspondence recently

introduced by the author in a related paper. In the process, we

also extend a classical result of Gordon–Litherland, establishing

an isomorphism between their pairing on a spanning surface and

the intersection form on a 4-manifold constructed as a double-

branched cover using that surface.

1. Introduction

P.G. Tait asserted in 1898 that all reduced alternating diagrams

of a given prime nonsplit link in S3minimize crossings, have equal

writhe, and are related by ﬂype moves (see Figure 1) [Ta1898]. The

ﬁrst proofs came almost a century later, and all involved the Jones

polynomial [Ka87, Mu87, Mu87ii, Th87, MT91, MT93]. In 2017,

Greene gave the ﬁrst purely geometric proof of part of the classical

Tait conjectures [Gr17], and in 2020, the author gave the ﬁrst purely

geometric proof of Tait’s ﬂyping conjecture [Ki21].

Recently, Boden, Chrisman, Karimi, and Sikora extended much of

this to alternating links in thickened surfaces. First, using general-

izations of the Kauﬀman bracket, Boden–Karimi–Sikora proved that

Tait’s ﬁrst two conjectures hold for alternating links in thickened sur-

faces [BK18, BKS19].1Second, Boden–Chrisman–Karimi extended

the Gordon–Litherland pairing to spanning surfaces in thickened sur-

faces [BCK21]. Third, Boden–Karimi applied this pairing to extend

Greene’s characterization of classical alternating links to links Lin

thickened surfaces Σ ×I, proving that Lbounds connected deﬁnite

surfaces of opposite signs if and only if Lis alternating and (Σ×I, L)

is nonstabilized [BK22].2

1Boden–Karimi proved Tait’s ﬁrst two conjectures for alternating links in

thickened surfaces, with a few extra conditions [BK18], and with Sikora they ex-

tended those results to adequate links and removed the extra conditions [BKS19].

2See §2.1 for deﬁnitions of stabilized,prime,locally prime,cellular,end-

essential,deﬁnite, and removably nugatory.

arXiv:2210.03720v2 [math.GT] 29 Aug 2024

2 THOMAS KINDRED

Figure 1. Aﬂype along an annulus A=νγ ⊂Σ.

The ﬁrst main result of this paper combines and adapts several of

these recent developments to prove that the ﬂyping theorem extends

to alternating links in (nonstabilized) thickened surfaces.

Theorem 3.5. Let D⊂Σbe a locally prime, cellular alternating

diagram of a link Lin a thickened surface Σ×I.Then any other

such diagram of Lis related to Dby ﬂypes on Σ.

The approach is parallel to that in [Ki21], and indeed most of

the arguments translate directly. For some, which we mark with the

symbol ^, the statements and proofs hold without further comment.

Appendix A lists pertinent cross-referencing information for these

and other results marked with the symbol . The upshot is a geo-

metric proof of Theorem 3.5 and other generalized Tait conjectures:

Theorem 3.3 (Part of Tait’s extended ﬁrst conjecture [BK18, BKS19]).

If D, D′⊂Σare alternating diagrams of a link L⊂Σ×I, neither

containing removable nugatory crossings, then Dand D′have the

same number of crossings.

Theorem 3.6 (Tait’s extended second conjecture [BK18, BKS19]).

All locally prime, cellular alternating diagrams of a given link L⊂

Σ×Ihave the same writhe.

Section 4 extends Theorems 3.3, 3.5, and 3.6 to virtual links:

Theorem 4.11. Any two locally prime, alternating virtual diagrams3

of a given virtual link Kare related by non-classical R-moves and

(classical) ﬂypes.4

To prove this, we establish a new diagrammatic analog to the corre-

spondence established by Kauﬀman, Kamada–Kamada, and Carter–

Kamada–Saito between virtual links, equivalence classes of abstract

links, and stable equivalence classes of links in thickened surfaces

[Ka98, KK00, CKS02]; also see [Ku03]. Given a virtual link diagram

3A virtual link diagram is alternating if its classical crossings alternate between

over and under.

4A (classical) ﬂype on a virtual link diagram appears as in Figure 1, where

T1contains no virtual crossings.

THE VIRTUAL FLYPING THEOREM 3

V, let [V] denote its equivalence class under virtual (non-classical)

R-moves. We show that such classes [V] correspond bijectively to ab-

stract link diagrams and thus to cellularly embedded link diagrams

on closed surfaces.

As corollaries, we extend Theorems 3.3 and 3.6 to virtual dia-

grams, and we observe that connect sum is not a well-deﬁned oper-

ation on virtual knots or links:

Theorem 4.12. All locally prime, alternating diagrams of a given

virtual link have the same crossing number and writhe.

Corollary 4.13. Given any two non-classical, locally prime, alter-

nating virtual links V1and V2, there are inﬁnitely many distinct vir-

tual links that decompose as a connect sum of V1and V2.

Before all this, in §2, we introduce the required background re-

garding links in thickened surfaces. Some of this reviews the existing

literature, and some of it is new.

2. Links and spanning surfaces in thickened surfaces

Convention 2.1. Throughout, Σ is a connected, closed, orientable

surface with genus g(Σ) >0.5We denote the intervals [−1,1] and

[0,1] by Iand I+, respectively. In Σ ×I, we identify Σ with Σ × {0}

and denote Σ × {±1}= Σ±. For a pair (Σ, L) or (Σ ×I, L), Lis a

link in Σ ×I, and for a pair (Σ, D), Dis a link diagram on Σ.

2.1. Alternating links in thickened surfaces. A pair (Σ, L) is

stabilized if, for some circle6γ⊂Σ, Lcan be isotoped so that it

intersects each component of (Σ ×I)\(γ×I) but not the annulus

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

γ×Iand attaching two 3-dimensional 2-handles in the natural way

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

Equivalently, (Σ, L) is nonstabilized if every diagram Dof Lon Σ is

cellularly embedded, meaning that Dcuts Σ into disks.

A pair (Σ, L) is split if Lhas a disconnected diagram on Σ. Note

that if (Σ, L) is split then it is also stabilized (as we assume that

Σ is connected). The converse is false. In fact, the number of split

components is an invariant of stable equivalence classes.

Kuperberg’s Theorem states that the stable equivalence class of

(Σ, L) contains a unique nonstabilized representative; this implies

that when (Σ, L) is nonsplit, (Σ, L) is nonstabilized if and only if Σ

has minimal genus in this stable equivalence class.

Theorem 2.2 (Theorem 1 of [Ku03]).If (Σ, L)and (Σ′×I, L′)are

stably equivalent and nonstabilized, then there is a pairwise homeo-

morphism (Σ ×I, L)→(Σ′×I, L′).

5[Ki22, Ki23] also allow Σ to be disconnected with components of any genus.

6We use “circle” as shorthand for “simple closed curve.”

4 THOMAS KINDRED

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

Σ×Iis always irreducible, since its universal cover is R2×R.7The

converse of this, too, is false,8due to the next observation, which

follows from a standard innermost circle argument:

Observation 2.3. If (Σi×I)\Liis irreducible for i= 1,2and

Σ=Σ1#γΣ2with L=L1⊔L2⊂Σ×I, where the annulus A=γ×I

separates L1from L2in Σ×I, then (Σ ×I)\Lis irreducible.

We call (Σ, D)cellular if Dcuts Σ into disks. Note:

Fact 2.4 ([Oz06, BK22]; Proposition 5.1 of [Ki22]).Suppose D⊂Σ

is a cellular alternating diagram of a link L⊂Σ×I. Then (Σ, D)is

checkerboard colorable, and Lis nullhomologous over Z/2.

We will use this result of Boden–Karimi and the generalization

that follows:

Fact 2.5 (Corollary 3.6 of [BK22]).If (Σ, L)has a cellular alternat-

ing diagram, then (Σ, L)is nonsplit and nonstabilized.

Corollary 2.6 (Corollary 2.4 of [Ki22]).Suppose (Σ, L)has an al-

ternating diagram D⊂Σ. Then (Σ, L)is nonsplit if and only if D

is connected, and (Σ, L)is nonstabilized if and only if Dis cellular.

Following [Ki22], we call a checkerboard colorable pair (Σ, D)

pairwise prime if any pairwise connect sum decomposition (Σ, D) =

(Σ1, D1)#(Σ2, D2) has (Σi, Di) = (S2,⃝) for either i= 1,2. Like-

wise, given (Σ, L) where Lis nullhomologous over Z/2, we call (Σ, L)

pairwise prime if every annular connect sum decomposition (Σ, L) =

(Σ1, L1)#(Σ2, L2) is trivial: (Σi, Li) = (S2,⃝) for either i= 1,2.9

10 Thus, such (Σ, L) is pairwise prime if and only if, whenever γ⊂Σ

is a separating curve and Lis isotoped to intersect the annulus γ×I

in two points, γbounds a disk X⊂Σ such that Lintersects X×I

in a single unknotted arc.

Howie-Purcell call (Σ, D)weakly prime if, for every pairwise con-

nect sum decomposition (Σ, D) = (Σ, D1)#(S2, D2), either D2=⃝

is the trivial diagram of the unknot or (Σ, D1) = (S2,⃝) [HP20];

following [Ki22], we call such Dlocally prime. We call (Σ, L)lo-

cally prime if, for every pairwise connect sum decomposition (Σ, L) =

7For more detail, see Proposition 12 of [BK22]; the proof cites [CSW14].

8If (Σi×I, Li) is nonsplit (implying that Σi×I\Liis irreducible) for i= 1,2,

then choose disks Xi⊂Σiwith (Xi×I)∩Li=∅and construct the connect sum

Σ = (Σ1\int(X1)) ∪(Σ2\int(X2)) = Σ1#Σ2. Let L=L1⊔L2⊂Σ×I. Then

(Σ, L) is split. Yet, (Σ ×I)\Lis irreducible by Observation 2.3.

9Annular connect sum (Σ1, L1)#(Σ2, L2) = (Σ, L) is a connect sum of sur-

faces, Σ1#Σ2= Σ, thickened up, which restricts to a connect sum of 1-manifolds

L1#L2=L. See [Ka98, Ma12, Ki22].

10The deﬁnitions of pairwise primeness are more complicated without the as-

sumptions related to checkerboard colorability; see [Ki22].

THE VIRTUAL FLYPING THEOREM 5

(Σ, L1)#(S2, L2), either L2=⃝is the unknot or (Σ, L1) = (S2,⃝)

[HP20].11

As in the classical case [Me84], certain diagrammatic conditions

constrain an alternating link Las one might wish:

Theorem 2.7 ([Oz06, BK22, Aetal19, Ki22]).If D⊂Σis a cellular

alternating diagram of a link L⊂Σ×I, then (i) Lis nonsplit, so

in particular, (Σ ×I)\Lis irreducible if g(Σ) >0; (ii) if (Σ, D)

is locally prime, then (Σ, L)is locally prime; and (iii) if (Σ, D)is

pairwise prime, then (Σ, L)is pairwise prime.

Part (i) was proven by Ozawa in [Oz06] and by Boden-Karimi

in [BK22]. Part (ii) was proven by Adams et al in [Aetal19] and

generalized by Howie-Purcell in [HP20]. Part (iii) is one of the main

results of [Ki22], which also gives new proofs of (i)-(ii).

2.1.1. End-essential spanning surfaces. Part (i) of Theorem 2.7 im-

plies that Lhas spanning surfaces: embedded, unoriented, compact

surfaces F⊂Σ×Iwith ∂F =L; while we do not require Fto be

connected, we do require that each component of Fhas nonempty

boundary. By deleting the interior of a regular neighborhood of L

from Fand Σ ×I, one may instead view Fas a properly embed-

ded surface in the link exterior (Σ ×I)\◦

νL1213 We take this view

throughout, except in Deﬁnition 2.8, Note 20, and §2.3.1.

If (Σ, D) is a cellular alternating diagram of (Σ, L), then it is

possible to orient each disk of Σ \Dso that, under the resulting

boundary orientation, over- and under-strands are oriented respec-

tively toward and away from crossings. Since Σ is orientable, these

orientations determine a checkerboard coloring of Σ\\D,14 i.e. a way

of shading the disks of Σ \\Dblack and white so that regions of the

same shade abut only at crossings.15 One can use this checkerboard

coloring to construct checkerboard surfaces Band Wfor L, where

Bprojects into the black regions, Wprojects into the white, and B

and Wintersect in vertical arcs which project to the the crossings

of D. The main result of [Ki23] is that these checkerboard surfaces

satisfy several convenient properties:

11A third notion of primeness for Don Σ also appears in the literature: Ozawa

calls (Σ, D)strongly prime if every circle on Σ (not necessarily separating) that

intersects Din two generic points also bounds a disk in Σ which contains no

crossings of D[Oz06].

12Throughout, given a manifold Xand a submanifold Y⊂X,νY denotes a

closed regular neighborhood of Yin X.

13We also assume that ∂F is transverse on ∂νL to each meridian, where a

meridian is the preimage of a point in Lunder the bundle map νL →L.

14For compact X, Y ⊂Σ×I,X\\Ydenotes the metric closure of X\Y; see

Note 7 of [Ki21] for a precise deﬁnition.

15Interestingly, cellular alternating link diagrams on nonorientable surfaces are

never checkerboard colorable.

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THEVIRTUALFLYPINGTHEOREMTHOMASKINDREDAbstract.Weextendtheflypingtheoremtoalternatinglinksinthickenedsurfacesandalternatingvirtuallinks.TheproofoftheformerresultusesworkofBoden–Karimitoadapttheauthor’sgeometricproofofTait’s1898flypingconjecture(firstprovedin1993byMenasco–Thistlethwaite),whiletheproof...

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