END-ESSENTIAL SPANNING SURFACES FOR LINKS IN THICKENED SURFACES THOMAS KINDRED

2025-04-29 0 0 2.83MB 15 页 10玖币

侵权投诉

END-ESSENTIAL SPANNING SURFACES FOR LINKS

IN THICKENED SURFACES

THOMAS KINDRED

Abstract. Let Dbe a cellular alternating link diagram on a

closed orientable surface Σ. We prove that if Dhas no remov-

able nugatory crossings then each checkerboard surface from

Dis π1-essential and contains no essential closed curve that is

∂-parallel in Σ ×I. Our chief motivation comes from techni-

cal aspects of a companion paper, where we prove that Tait’s

ﬂyping conjecture holds for alternating virtual links. We also

describe possible applications via Turaev surfaces.

1. Introduction

Let Σ be a closed orientable surface, not necessarily connected or

of positive genus, and let D⊂Σ be an alternating diagram of a

link L⊂Σ×Isuch that Dcuts Σ into disks; such Dis said to be

cellular alternating. Then the disks of Σ\\Dadmit a checkerboard

coloring, from which one can construct the checkerboard surfaces B

and Wof D.1

These are spanning surfaces for L: embedded compact surfaces

in int(Σ ×I) with no closed components and whose boundary is L.

We call a spanning surface Ffor Lend-essential if it is π1-injective

and ∂-incompressible and contains no essential closed curve which is

parallel in (Σ ×I)\\Fto ∂(Σ ×I). See Deﬁnition 1.6 for details.

A crossing cin Dis removably nugatory if there is a disk X⊂Σ

such that ∂X ∩D={c}. In that case, one can remove cfrom Dby

using a ﬂype to move cacross the part of Din Xand then using

a Reidemeister 1 move undo the resulting monogon. If Dhas a

removable nugatory crossing, then Bor Wis ∂-compressible. Our

main result is the following strong converse of this fact:

Theorem 1.1. If D⊂Σis a cellular alternating link diagram with-

out removable nugatory crossings, then both checkerboard surfaces

from Dare end-essential.

1We denote I= [−1,1]. In Σ ×I, we identify Σ with Σ × {0}and denote

Σ×{±1}= Σ±. Also, X\\Ydenotes Xcut along Y. Formally, this is the metric

closure of X\Y. It is homeomorphic to X\◦

νY , but with extra structure from

Yencoded in its boundary.

arXiv:2210.03218v2 [math.GT] 29 Aug 2024

2 THOMAS KINDRED

Figure 1. The surface F(left) is end-essential, as is

its mirror image F′, but not F ♮F ′(right).

Our proof strategy for Theorem 1.1 is to prove the result with

an extra (weak) primeness assumption on Dand then extend via

connect sum. We note that in some situations, the conclusion of the

theorem follows from work of Ozawa [Oz06] and Howie [Ho15].

If γ⊂Σ is a separating curve such that the annulus A=γ×I

intersects the link L⊂Σ×Itransversally in two points, then cutting

Σ×Ialong Aand gluing on (in the natural way) two 3-dimensional

2-handles, each containing a properly embedded arc, decomposes

(Σ, L) as an annular connect sum (Σ, L) = (Σ1, L1)#γ(Σ2, L2).

The factors (Σi, Li) are uniquely determined by (Σ, L) and γup to

pairwise homeomorphism (generally, however, the factors (Σ1, Li) do

not determine (Σ, L) uniquely). There is also a diagrammatic version

of annular connect sum. See [Ki22b] for details.

With this setup, if moreover Fspans Land |A⋔F|= 1,2

Observation 1.2. Even if Fi⊂Σi×Iis end-essential for each

i= 1,2, the surface F1♮F2⊂(Σ1#Σ2)×Ineed not be end-essential.

Indeed, consider the example shown in Figure 1. The surface

F⊂T2×Ishown left is end-essential (this will follow from Theorem

1.4). So too is its mirror image F′⊂T2×I. Yet, as shown right

in the ﬁgure, F ♮F ′is not end-essential in (T2#T2)×I: the white

curve on F ♮F ′is parallel to the dotted curve on Σ+. We note that

this behavior is related to the following phenomenon in the classical

setting:

Observation 1.3. Even if Fi⊂S3is π1-injective for i= 1,2, the

surface F1♮F2⊂S3need not be π1-injective.

Indeed, if Mand M′are M¨obius bands, each spanning an unknot

in S2×I, then both Mand M′are π1-injective (but ∂-compressible;

in fact, they are the only connected spanning surfaces in S3which are

π1-injective and ∂-compressible), and yet M♮M′is not π1-injective.

2Here and throughout, |X|denotes the number of components of X. The

notations |A∩F|and |A⋔F|carry the same meaning; we use the latter notation

if we wish to emphasize or clarify that Aand Fare transverse.

END-ESSENTIAL SPANNING SURFACES 3

The key to dealing with the complication presented by Observa-

tion 1.2 is to distinguish between annular connect sums (Σ, L) =

(Σ, L1)#(Σ, L2) in general and those which are local in the sense

that one of Σi=S2. Indeed, this distinction is at the heart of

[Ki22b].

Following Howie–Purcell [HP20], we call a pair (D, Σ) weakly

prime if Dis nontrivial and, for any local connect sum decomposi-

tion (Σ, D) = (Σ, D1)#(S2, D2), either D2=⃝is the trivial diagram

of the unknot, or (Σ, D1) = (S2,⃝) [HP20]. Note that no weakly

prime, cellular alternating link diagram with more than one crossing

has any removable nugatory crossings.

Theorem 1.1 will follow from the following two results:

Theorem 1.4. If D⊂Σis a weakly prime, cellular alternating link

diagram with more than one crossing, then both checkerboard surfaces

from Dare end-essential.

Proposition 1.5. Suppose (Σ, L) = (Σ1, L1)#(Σ2, L2). If F=

F1♮F2spans L, where each Fiis a π1-essential spanning surface for

Li, then Fis π1-essential. If moreover Σ2=S2and F1is end-

essential, then Fis also end-essential.

Before presenting the proofs, we give a more precise deﬁnition of

end-essentiality. Note that these properties all concern curves and

arcs in Fwhich may self-intersect. There are alternative notions

which do not allow such self-intersections; those notions are some-

times referred to as geometric (since they indicate whether or not cer-

tain types of surgery moves are possible on F) and these as algebraic

(due to the equivalence between incompressibility and π1-injectivity:

see the next remark below) [Ki24].

Deﬁnition 1.6. Let Fbe a spanning surface for a link Lin Σ ×I,

and write MF= (Σ×I)\\F. Write hF:MF→Σ×Ifor the quotient

map that reglues corresponding pairs of points from int(F) in ∂MF,

and denote e

L=hF−1(L), f

Σ±=h−1

F(Σ±), and e

F=h−1

F(int(F)) =

∂MF\(f

Σ±∪e

L), so that hFrestricts to a homeomorphism MF\e

F→

(Σ ×I)\int(F) and to a 2:1 covering map e

F→int(F). Then we say

that Fis:

(a) incompressible if any circle3γ⊂e

Fthat bounds a disk

X⊂MFalso bounds a disk in e

F. In that case, X=hF(X)

is called a fake compressing disk for F; if γdoes not bound

a disk in e

Fthen Xis a compressing disk for F.

(b) end-incompressible if any circle γ⊂e

Fthat is parallel

through an annulus e

Ain MFto f

Σ±bounds a disk in e

F. In

3We use “circle” as shorthand for “simple closed curve.” A circle in a surface

is essential if it does not bound a disk in that surface.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

END-ESSENTIALSPANNINGSURFACESFORLINKSINTHICKENEDSURFACESTHOMASKINDREDAbstract.LetDbeacellularalternatinglinkdiagramonaclosedorientablesurfaceΣ.WeprovethatifDhasnoremov-ablenugatorycrossingstheneachcheckerboardsurfacefromDisπ1-essentialandcontainsnoessentialclosedcurvethatis∂-parallelinΣ×I.Ourchiefmo...

展开>> 收起<<

END-ESSENTIAL SPANNING SURFACES FOR LINKS IN THICKENED SURFACES THOMAS KINDRED.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

END-ESSENTIAL SPANNING SURFACES FOR LINKS IN THICKENED SURFACES THOMAS KINDRED

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: