SELF-DUAL MAPS III PROJECTIVE LINKS LUIS MONTEJANO1 JORGE L. RAM IREZ ALFONS IN2 AND IVAN RASSKIN Abstract. In this paper we present necessary and sucient combinatorial conditions for

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SELF-DUAL MAPS III: PROJECTIVE LINKS

LUIS MONTEJANO1, JORGE L. RAM´

IREZ ALFONS´

IN2, AND IVAN RASSKIN

Abstract. In this paper, we present necessary and suﬃcient combinatorial conditions for

a link to be projective, that is, a link in RP3. This characterization is closely related to

the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the

notion of symmetric cycle, an interesting issue arising in projective links leading us to an

easy condition to prevent a projective link to be alternating.

1. Introduction

This paper is a continuation of the work [7] where the notions of antipodally self-dual and

antipodally symmetric maps were studied and the work [8] where these notions were applied

to investigate questions concerning symmetry and amphicharility of links. It turns out that

the above notions ﬁt also nicely in order to understand better projective links, that is, links

in RP3.

Projective links have been studied in diﬀerent contexts : in connection with the twisted

Alexander polynomial [5] and also in relation with unknotting issues [9] and possibly with

3-manifolds [10, Chapter IX]. In [2], Drobotukhina presented the analogue of the Jones

polynomial for projective links and then used it in [3] to give a classiﬁcation of projective

links with at most six crossings.

In this paper, we present necessary and suﬃcient combinatorial conditions for a link to be

projective. The latter was done by using a characterization of projective links in terms of

some special embeddings in 3-space invariant under negative inversions.

In the next section we overview some basic notions of knots, projective links and maps

needed for the rest of the paper. In Section 3, we present our new approach and the above

mentioned characterizations. In Section 4, we discuss the notion of symmetric cycle, an

interesting issue arising in projective links leading us to an easy condition to prevent a

projective link to be alternating.

2. Knots and maps : preliminaries

2.1. Knots background. We refer the reader to [1] of [6] for standard background on knot

theory.

Alink with kcomponents consists of kdisjoint simple closed curves (S1) in R3. A knot

Kis a link with one component. A link diagram D(L) of a link Lis a regular projection of

Linto R2in such a way that the projection of each component is smooth and at most two

curves intersect at any point. At each crossing point of the link diagram the curve which

2010 Mathematics Subject Classiﬁcation. Primary 57M15, 57K10.

Key words and phrases. Self-dual Maps, Projective Links.

1Partially supported by CONACyT 166306 and PAPIIT-UNAM IN112614.

2Partially supported by grant PICS07848 and INSMI-CNRS.

arXiv:2210.04053v1 [math.GT] 8 Oct 2022

goes over the other is speciﬁed. A shadow of a link diagram Dis a 4-regular graph if the

over/under passes of Dare ignored. Since the shadow is Eulerian (4-regular) then its faces

can be 2-colored, say with colors black and white. We thus have that each vertex is incident

to 4 faces alternatively colored around the vertex, see Figure 1.

Figure 1. (From left to right) A diagram of the Trefoil, its shadow with a

2-colored faces (vertices on white crossed circles), corresponding Black graph

(bold edges and black circles) and White graph (dotted edges and white cir-

cles).

Given such a coloring, we can deﬁne two graphs, one on the faces of each color. Let

BDdenote the graph with black faces as its vertices and two vertices are joined if the

corresponding faces share a vertex (BDis called the checkerboard graph of D). We deﬁne

the graph WDon the white faces of the shadow analogously. We notice that med(WD) is

exactly the shadow of Dand notice that med(BD) and med(WD) are the same.

An edge-signed planar graph, denoted by (G, SE), is a planar graph Gequipped with a

signature on its edges SE:E→ {+,−}. We will denote by −SEthe signature of Gsatisfying

−SE(e) = −(SE(e)) for every e∈E. We write S+

E(resp. S−

E) when all the signs of SEare +

(resp. −). Given a crossing of the link diagram we sign positive or negative according to the

left-over-right and right-over-left rules from point of view of black around the crossing, see

Figure 2 (Left). The latter induce an opposite signing on each crossing by the same rules

but now from point of view of white around the crossing, see Figure 2 (Right).

Figure 2. (Left) Left-over-right rule from black point of view. (Right) Right-

over-left rule from white point of view.

If the crossing is positive, relative to the black faces, then the corresponding edge is

declared to be positive in BDand negative in WD. Therefore, in this fashion, a link diagram

Ddetermine a dual pair of signed planar graphs (BD, SE) and (WD,−SE) where the signs

on edges are swapped on moving to the dual.

Remark 1. A link diagram can be uniquely recovered from either (BD, SE)or (WD,−SE).

We thus have that given an edge-signed planar graph (G, SE), we can associate to it (in a

canonical way) a link diagram D(L) such that (BD, SE) (and (WD,−SE)) gives (G, SE). The

unsigned graph Gis called the Tait graph of the link Lwith diagram D. The construction

is easy, we just consider the med(H) with signatures on the vertices (induced by the edge-

signature SEof G). The desired diagram, denoted by D(G, SE), is obtained by determining

the under/over pass at each crossing according to Left-over-right (or Right-over-left) rule

associated to the sign of the corresponding edge of G, see Figure 3.

Figure 3. (From left to right) diagram Dof the Hopf link, denoted by 21,

signed Black graph (BD, SE) and signed White graph (WD,−SE).

2.2. Diagrams for links in RP3.Aprojective n-link is the image of a smooth embedding

of ndisjoint copies of S1in RP3.

The 3-dimension real projective space RP3can be deﬁned as the sphere S3with identiﬁed

opposite points. Since S3consists of two half-spheres then we can restrict ourselves to the

upper hemisphere of S3and merely identify antipodal points on the bounding equator. Now,

since each half-sphere is homeomorphic to the ball B3then have that RP3can be obtained

by identifying diametrically opposite points of the boundary S2=∂B3.

Consequently, any link Lin RP3can be deﬁned as a set of closed curves and arcs in B3

such that the set of endpoints of arcs lies in ∂B3. We say that the link Lin RP3is lifted to

L0in B3. Up to isotopy, we can assume that the images of the poles of B3do not belong to

L0.

A projective link Lcan be represented by projective diagrams that diﬀer from usuals

link diagrams in R3in that they are given not on the plane but in a closed 2-disc, and

the endpoints of arcs at the boundary of the 2-disc are divided into pairs of diametrically

opposite points.

More precisely, let π:L0→δbe the projection of L0to the equatorial disc ∆ ⊂B3deﬁned

π(x) = Cx∩∆

where Cx⊂B3is the semicircle passing through xand the poles of the ball. Up to a small

isotopy, we can assume that Lsatisﬁes the following general position properties:

a) the image π(L0) does not contain any cusp, tangency point or triple point,

b) L0is a smooth submanifold of B3intersecting transversally the boundary sphere ∂B3

c) there do not exist 2 points L0∩∂B3projecting to the same point under π.

We thus have that a link in RP3give rise a projective diagram by projecting to the

equatorial disc B2. Conversely, given a projective diagram regarding B2as the equatorial

disc of such a representation of RP3. At each crossing, the upper arc is pull up (and the

lower one is pull down) in order to obtain a nonintersecting curve lying inside B3. The

identiﬁcation of antipodal points lying on the boundary sphere ∂B3gives rise to a link in

RP3.

a b

Figure 4. Nonintersecting curves inside B3(induced by a link in RP3) and

its projective diagram in RP2.

In Table 1 we present the ﬁrs 14 nontrivial projective links among the 111 projective links

with at most 6 crossings appearing in [3, Table of links in RP3, page 102].

Table 1. First nontrivial projective links.

We notice that any projective diagram in RP2arises a projective link. However, it may

happens that two such diagrams lead to the same projective link.

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SELF-DUALMAPSIII:PROJECTIVELINKSLUISMONTEJANO1,JORGEL.RAMIREZALFONSIN2,ANDIVANRASSKINAbstract.Inthispaper,wepresentnecessaryandsucientcombinatorialconditionsforalinktobeprojective,thatis,alinkinRP3.Thischaracterizationiscloselyrelatedtothenotionsofantipodallyself-dualandantipodallysymmetricmaps.W...

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