AN ANALOG OF THE KAUFFMAN BRACKET POLYNOMIAL FOR KNOTS IN THE NON-ORIENTABLE THICKENING OF A NON-ORIENTABLE SURFACE_2

2025-04-30 0 0 550.15KB 19 页 10玖币

侵权投诉

AN ANALOG OF THE KAUFFMAN BRACKET POLYNOMIAL

FOR KNOTS IN THE NON-ORIENTABLE THICKENING OF A

NON-ORIENTABLE SURFACE

VLADIMIR TARKAEV

Abstract. We study pseudo-classical knots in the non-orientable thickening

of a non-orientable surface, speciﬁcally knots that are orientation-preserving

paths in a non-orientable 3-manifold of the form (non-orientable surface) ×

[0,1]. For these knots, we propose an analog of the Kauﬀman bracket polyno-

mial. The construction of this polynomial closely mirrors the classical version,

with key diﬀerences in the deﬁnitions of the sign of a crossing and the pos-

itive/negative smoothing of a crossing. We prove that this polynomial is an

isotopy invariant of pseudo-classical knots and demonstrate that it is indepen-

dent of the classical Kauﬀman bracket polynomial for knots in the thickened

orientable surface, which is the orientable double cover of the non-orientable

surface under consideration.

Key words: knots in non-orientable manifold, knots in thickened surface,

pseudo-classical knots, Kauﬀman bracket polynomial

Introduction

Initially, the theory of knots studied knots in the 3-sphere, but now many other

kinds of object are also considered in the theory. In the context of the present paper,

ﬁrst of all we need to mention knots in a thickened surface, that is in an orientable

3-manifold of the form (orientable surface) ×[0,1]. Some authors consider the case

of knots in a thickened non-orientable surface ( [1], [2], [6], [11]) where the thickening

is understood as an orientable 3-manifold equipped with an I-bundle structure over

a surface which is not necessarily orientable (here Idenotes a segment). Note that

in all these cases the thickening is assumed to be orientable independently whether

the base surface is orientable or not.

Knots in non-orientable 3-manifolds are also studied (see, for example, [13], [14])

but, unfortunately, the subject is not very popular. Maybe, one of the reasons is

that some important constructions cannot be transferred straightforwardly from

an orientable to a non-orientable case. In particular, that is so for the Kauﬀ-

man bracket polynomial ( [7], [8]) and many other polynomial knot invariants that

explicitly use in their construction the notion of the sign of a crossing (see, for

example, [5], [9], [12], [10]). In the classical situation, the sign of a crossing xof

an oriented diagram Don a surface Σ is deﬁned to be 1 (resp. −1) if a pair (a

positive tangent vector of overgoing branch at x, a positive tangent vector of un-

dergoing branch at x) is positive (resp. negative) basis in the tangent space of

Σ at x. Clearly, the deﬁnition cannot be used if Σ is non-orientable. One more

example is the notion of positive and negative smoothing of a crossing (in other

terms, the smoothing of the type Aand of the type B) which play a key role in

The work is supported by RSF (grant number 23-21-10014).

arXiv:2210.00540v4 [math.GT] 3 Sep 2024

2 VLADIMIR TARKAEV

the construction of the Kauﬀman bracket polynomial. The deﬁnition of the latter

notion uses concepts of “right” and “left” which have no sense on a non-orientable

surface. Note that aforementioned notions can be reformulated so that they have

sense in the case of the orientable thickening of a non-orientable surface. However,

the approach allowing to do this cannot be used in the case of the non-orientable

thickening of a non-orientable surface.

In the present paper, we study knots in the non-orientable thickening of a non-

orientable surface, that is a non-orientable 3-manifold of the form Σ×[0,1], where Σ

is a non-orientable surface with (maybe) non-empty boundary. We propose an ap-

proach that in some particular case gives correct deﬁnitions of the sign of a crossing

and of two types of a smoothing. The knots we mean (we call them pseudo-classical)

are those which are orientation-preserving paths in the manifold. Two aforemen-

tioned deﬁnitions allow deﬁning an analog of the Kauﬀman bracket polynomial

for pseudo-classical knots. The construction we use word-for-word coincides with

the classical one, the speciﬁcity of the non-orientable thickening is hidden in using

deﬁnition of positive and negative smoothing. Our proof of the invariance of the

polynomial (we denote it by J) is close to the classical one but do not coincide

with the latter. Finally, we compare the polynomial Jwith its natural competi-

tor — with the Kauﬀman bracket polynomial for knots in the thickened orientable

surface, which is the orientable double cover of the non-orientable surface under

consideration. We demonstrate that none of these invariants is a consequence of

the other. Namely, we consider two pairs of knots in the non-orientable thickening

of the Klein bottle so that knots forming the ﬁrst pair have coinciding polynomi-

als Jwhile 2-component links in the thickened torus those are their double covers

have distinct Kauﬀman bracket polynomials. Knots forming the second pair, on

the contrary, have distinct polynomials Jwhile their double covers have coinciding

Kauﬀman bracket polynomials.

The paper is organized as follows. In Section 1, we give some deﬁnitions, includ-

ing a new deﬁnition of the sign of a crossing. In Section 2, we deﬁne two types of

smoothing. Then, we deﬁne the polynomial Jand prove its invariance. In the end

of this section, we brieﬂy discuss a generalization of the polynomial analogous to the

homological version of the Kauﬀman bracket polynomial for knots in a thickened

orientable surface [3]. In Section 3, we compare the polynomial Jand the Kauﬀman

bracket polynomial for orientable double cover. In particular, in Section 3.2, we

explicitly compute the polynomial Jfor two knots in the non-orientable thickening

of the Klein bottle. In Appendix, we give source data needed for computing the val-

ues of the Kauﬀman bracket polynomial in the format acceptable by the computer

program “3–manifold recognizer”.

1. Definitions

1.1. Knots and diagrams. Throughout this paper Σ denotes a non-orientable

surface with (maybe) non-empty boundary and ˆ

Σ denotes the non-orientable man-

ifold of the form Σ ×[0,1]with ﬁxed structure of the direct product. The latter

condition becomes crucial for us in the case of surface with non-empty bound-

ary when there are aforementioned structures of the corresponding manifold with

homeomorphic but non-isotopic base surfaces.

A KAUFFMAN BRACKET POLYNOMIAL FOR THE NON-ORIENTABLE THICKENING OF A NON-ORIENTABLE SURFACE3

Knots in ˆ

Σ and their diagrams on Σ can be deﬁned by analogy with the case of

knots in a thickened orientable surface (below, we will refer to the latter case as

the “classical case”). Knots in ˆ

Σ will be considered up to ambient isotopy.

In our case, two diagrams on Σ represent the same knot if and only if they are

connected by a ﬁnite sequence of Reidemeister moves R1, R2, R3(which are just the

same as classical ones) and ambient isotopies. indeed, let Σ be represented by poly-

gon of which sides are assumed to be identiﬁed (maybe) with a twist and a diagram

is drawn inside the polygon. If we isotope the knot in question inside the direct

product of the polygon and the segment, then we can refer to the corresponding

classical fact. Otherwise, it is necessary to consider a few auxiliary moves. These

moves describe how we push arcs and crossings of the diagrams through the bound-

ary of the polygon. In the case of knots in orientable thickening of non-orientable

surface (see, for example, [2], [1]), some of these moves permute overgoing and un-

dergoing branches at pushing crossing. In our case, ˆ

Σ is the direct product hence

we have no moves permuting overgoing and undergoing branches, i.e., all auxiliary

moves represent ambient isotopies of a diagram. Therefore, in proofs below, we can

restrict ourselves to Reidemeister moves R1, R2, R3.

1.2. Pseudo-classical knots. A knot K⊂ˆ

Σ is called an pseudo-classical if it is

an orientation-preserving path in ˆ

Σ. In other words, a knot in ˆ

Σ is pseudo-classical

if its regular neighborhood in the manifold is homeomorphic to the solid torus.

Clearly, in a non-orientable manifolds not all knots have the property since, in

this case, there are knots whose regular neighborhood is homeomorphic to the solid

Klein bottle. However, the latter kind of knots is out of consideration in the present

paper.

Let Kbe a pseudo-classical oriented knot represented by a diagram D⊂Σ. Since

the knot is pseudo-classical, the diagram (or, more precisely, the projection of K

onto Σ) viewed as a closed directed path on the surface is an orientation-preserving

path. Hence, 2-cabling D2⊂Σ of Dis a diagram of a 2-component link (recall that

2-cabling of a diagram is, informally speaking, the same diagram drawn by doubled-

line with preserving over/under-information in all crossings). Note again that this

is not the case for an arbitrary diagram on Σ. This is because Σ is assumed to be

non-orientable hence there are knot diagrams on the surface, those are orientation-

reversing paths. Hence, their 2-cabling represents not a 2-component link but a

knot which goes along the given knot twice. In the latter case, 2-cabling bounds

the M¨obius strip (going across itself near each crossing of the given diagram) while

2-cabling of a pseudo-classical knot bounds an annulus (having self-intersections).

In the case of an oriented surface, one can speak of left and right components of

2-cabling of an oriented knot diagram. Since Σ is non-orientable, we do not have

the concepts of left and right. However, we will use the terms “left component”

and “right component” of D2keeping in mind that in our situation this is nothing

more than a convenient notation. A choice (which component is chosen as the left

and which is chosen as the right) will be called the labeling of D2. The components

will be denoted by L(D)and R(D), respectively. Below, L(D)and R(D)are

assumed to be oriented so that their orientations are agreed with the orientation of

D. Sometimes we will rename R(D)into L(D)and vice versa, the transformation

will be called the relabeling of D2.

A single crossing of Dbecomes in D2a pattern of 4 crossings (see Fig. 1). There

are two distinguished crossings in each of these patterns: the ﬁrst one in which

4 VLADIMIR TARKAEV

inp(x)

out(x)

R L

(a)

inp(x)

out(x)

R L

(b)

inp(x)

out(x)

R R

(c)

inp(x)

out(x)

L R

(d)

Figure 1. Segments belong to D, R(D)and L(D)are drawn by

solid, dashed and dotted lines, respectively.

components of D2come into the pattern and the second one in which they go out.

These crossings are called the input (resp. output) crossing for the crossing xand

they are denoted by inp(x)(resp. out(x)), where x∈#Dis that crossing of D

to which the pattern corresponds. Here and below we denote by #Dthe set of

crossings of a diagram D.

Note if the surface under consideration is orientable then input and output cross-

ings are necessarily intersections of distinct components of 2-cabling while in non-

orientable case the situation when input and output crossings are a self-crossing of

the same component of D2may occur.

1.3. The sign of a crossing. Recall that in the classical situation the sign of a

crossing xof an oriented diagram Don an oriented surface Fis deﬁned to be 1

(resp. −1) if the pair (t1, t2)is a positive (resp. negative) basis in the tangent space

of Fat x, where t1and t2are, respectively, a positive tangent vector of overgoing

and undergoing branches of Dat x. Clearly, the deﬁnition cannot be used if the

surface is non-orientable. In the following deﬁnition, we use a labeling of 2-cabling

of the diagram under consideration instead of the orientation of the surface in which

the diagram lies.

Deﬁnition 1. Let D⊂Σand D2be a diagram of an oriented pseudo-classical

knot and its 2-cabling. The sign of a crossing x∈#Dis deﬁned to be the mapping

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：

ANANALOGOFTHEKAUFFMANBRACKETPOLYNOMIALFORKNOTSINTHENON-ORIENTABLETHICKENINGOFANON-ORIENTABLESURFACEVLADIMIRTARKAEVAbstract.Westudypseudo-classicalknotsinthenon-orientablethickeningofanon-orientablesurface,specificallyknotsthatareorientation-preservingpathsinanon-orientable3-manifoldoftheform(non-ori...

展开>> 收起<<

AN ANALOG OF THE KAUFFMAN BRACKET POLYNOMIAL FOR KNOTS IN THE NON-ORIENTABLE THICKENING OF A NON-ORIENTABLE SURFACE_2.pdf

共19页,预览4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！

AN ANALOG OF THE KAUFFMAN BRACKET POLYNOMIAL FOR KNOTS IN THE NON-ORIENTABLE THICKENING OF A NON-ORIENTABLE SURFACE_2

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: