On the Seifert fibered space link group

On the Seifert fibered space link group

Boštjan Gabrovšek    Enrico Manfredi
Abstract

We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link. Mathematics Subject Classification 2010: Primary 57M27; Secondary 57M05.
Keywords: knots, links, Seifert fibered spaces, knot group, first homology group, twisted Alexander polynomial.

1 Introduction

A Seifert fibered space can be constructed from an -bundle over a surface , with some possible disjoint rational -surgeries performed along the fibers , . Here the coefficients and are two coprime integers where . The fibers along which the surgeries are performed are called exceptional fibers, all other fibers are called ordinary. If the base space is compact then is compact and the number of exceptional fibers is finite [10].

Links in Seifert fibered spaces are embeddings of several disjoint copies of circles . As for links in the -sphere, we may want to find suitable ways to represent them and we may want to find invariants to distinguish them.

One of the main reason to be interested in links in Seifert fibered spaces is the computation of Skein Modules of these particular -manifolds. Skein modules are not only important invariants of -manifolds, but are also useful for topological quantum field theories.

Links in -bundles are best described by arrow diagrams introduced by Mroczkowski in [6]. Since Seifert fibered spaces resemble -bundles, we use these types of diagrams as in [7]. In Section 2 we present a generalized planar arrow diagram for a link in a Seifert fibered space, together with a list of ten generalized Reidemeister moves that satisfy the request that two link diagrams represent the same link up to ambient isotopy if and only if there exists a finite sequence of generalized Reidemeister moves between the two diagrams.

On the other hand we can consider studying links only up to difffeomorphism, i.e. we say that two links and are diffeo-equivalent if and only if there exists a self-diffeomorphism such that . This condition is weaker than ambient isotopy and we explore it in Section 3 by providing a method of calculating the group of a link which is able to detect diffeo-inequivalent links.

In Section 4, the link group is abelianized in order to obtain the first homology group of the complement. In some cases it is possible to recover it directly from the diagram, using the homology class of the link components. With some assumption on the manifold and on the link, it is possible to understand the behaviour of the rank and of the torsion of the homology.

In Section 5 we exploit the group presentation and the homology characterization to compute through Fox calculus a class of twisted Alexander polynomials, corresponding to a particular 1-dimensional representation of the link group. This particular class contains the usual Alexander polynomial and a family of twisted polynomials that is able to keep track of the torsion of the group; for example the polynomials of this family becomes zero on local links. Moreover, as expected, the polynomials split under connected sum of links. At last, in Section 6, an example illustrates all the machinery developed.

2 Diagrams of links in Seifert fibered spaces

In order to define diagrams of knots inside Seifert fibered spaces, we need to explicit their construction.

Let be a compact surface and let be the fundamental polygon of where is the equivalence relation that identifies the points on the boundary of the fundamental polygon as depicted on Figure 1. Denote the edges by or . If is an orientable genus surface, we have the identification of the edges (Figure 1(a)), if is a sphere we have the identification (Figure 1(b)), and if is a non-orientable genus surface, we have the identification (Figure 1(c)).

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Figure 1: Fundamental polygons.

An arbitrary compact Seifert fibered space can be constructed from as follows. Take and by glueing each to , , we get the trivial circle bundle . Since is a disk, we can orient all the fibers coherently. If two oriented edges and are identified in , in order to get , we can identify the tori and in two essentially different ways: is glued to by identity or by a reflection on the component. According to this, we assign to each edge the sign and to each edge the sign . In both cases is chosen if the identification is made by the identity and otherwise.

After the above identifications the resulting space is just a compact -bundle over . We can get an arbitrary Seifert fibered space by performing -surgeries along a disjoint fibers , for , where and are again two coprime integers, 111If we refer to Seifert’s original paper [10], Seifert fibered spaces are usually denoted by ; in our case we use the assumption , which is not restrictive, since, if we add a surgery with coefficients , we get the same result..

We remark that for an orientable Seifert fibered space , the base space does not need to be oriented, but the values (and ) are determined. In the case is orientable, and in the case in non-orientable for all .

If a link lies in a thickened surface , the diagram of is just the regular projection of onto along with the information of under- and overcrossings with respect to the projection. If we present by its fundamental polygon, we obtain a set of curves in . By glueing to we get a -bundle over . We now map a link to by the induced projection and keep track of where passes the section by decorating the diagram with an arrow at the passage point, where the arrow indicates the direction we should travel along in order to cross and emerge from (Figure 2). Let us remark that this induced projection agrees with the fibration in the case we do not have any exceptional fibers in .

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Figure 2: Constructing an arrow diagram.

Since the projection maps an exceptional fiber to a point in the base space, it is enough to specify the image of each exceptional fiber in , which is done by placing a point on decorated by the surgery coefficient of the fiber (Figure 3).

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Figure 3: A diagram of a link (a) and its projection (b) in a Seifert fibered space with an orientable genus 2 base space and , and two exceptional points with coefficients and .

We complete this section by providing a list of generalized Reidemeister moves associated to the arrow diagrams. In the interior of and outside of exceptional fibers we have three classical moves Reidemeister moves and two moves and involving arrows [6, 7, 8](Figure 4).

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Figure 4: Classical Reidemesiter moves and two “arrow” moves and .

Generalized Reidemeister moves act across edges in (Figure 5). The move corresponds to pushing an arc over an edge [3, 6, 7], corresponds to pushing a crossing over the edge and comes in two variants: in the case is orientable and in the case is non-orientable [6, 7, 3]. The move corresponds to pushing an arrow over an edge, where the sign denotes the sign associated to the edge [6, 7]. The move corresponds to pushing an arc over the base point of and is similar to the move in [3]. The move comes in three flavours: if is a orientable genus surface we have , if is the 2-sphere we have , and if is a non-orientable surface we have the move . Figure 6 shows the geometrical interpretation of in the case of a double torus.

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Figure 5: Additional Reidemeister moves.
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Figure 6: Visualization of the move .

Diagrams with exceptional points are equipped with an additional “slide” move (also known as the band move) that corresponds to sliding an arc over the exceptional point in a diagram, i.e. sliding an arc of a link over the meridional disk of the solid torus which is attached when performing the -surgery. The move consists of going times around the exceptional point and adding arrows uniformly on every angle as shown on Figure 7, see [7] for more details.

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Figure 7: The slide move over an exceptional point.

3 The link group

If the Seifert fibered space is described as an -bundle over the surface with possible rational surgeries and a link in is described by a generalized arrow diagram as in the previous section, then we can find a presentation for the group of the link. Before the description, we recall a standard presentation for the group of the Seifert fibered space itself, following [9].

The group of the Seifert fibered space

If is orientable, the fundamental group has the following presentation:

(1)

For non-orientable the presentation is as follows:

(2)

Where is the genus of the underlying basis , the integer is the number of surgeries performed with indexes along the -fiber , for every , the generators (resp. for the unorientable case) are the standard generators of while and are chosen according to the glueing orientation of the lateral surface of and , respectively; the last generator represents the fiber.

The group of the link

Consider an arrow diagram of a link in a Seifert fibered space with base surface and fundamental polygon . Fix an orientation on the link and on its arrow diagram.

The overpasses of the link may encounter the boundary of : when this happens, we index this boundary points with the following rule. Fix one of the corners of as a base point ; we may assume that it is the left corner of the edge as in Figure 8. Starting from the base point and going counterclockwise on the boundary of the diagram, we index by only the points on the edges oriented according to that direction. The edges with the opposite orientation have the points of the links labeled by , so that and are identified.

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Figure 8: Reading the group generators on the arrow diagram.

Each time an arrow occurs, we use the convention that a new overpass begins. Moreover, we should assume that no overpass both starts and ends on the boundary or on an arrow. If that were the case, perform an move on the overpass.

For the presentation of the group of the link, we use the generators and and the indexes and described for the fundamental group of in the paragraph above, moreover, we add the generators and relations described below.

We associate to each overpass a loop , that is oriented by the left hand rule, according to the orientation of the overpass. The indexation of the generators associated to the overpasses should respect the following rule: are the generators of those overpasses that end on the boundary points , the generators correspond to , the generators correspond to the overpasses before the arrows (considering the arrow orientation), the generators correspond to the overpasses after the arrows and finally are the remaining ones. Refer to Figure 9 for an example. For the loops with and we should add a sign if the overpass, according to the orientation, enters from the boundary, otherwise. Clearly and .

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Figure 9: The loops representing the group generators.

For each exceptional fiber , , let denote the generator associated to the fiber and fix a path on the diagram, connecting to the fiber point. Also for each arrow we fix a path on the diagram connecting with the arrow. These paths can not intersect each other and must follow the index order (before the fiber ones and then the arrow ones), as shown in Figure 8; we may also require that they are in general position with respect to the link projection. If we read the overpass generators that we meet along the paths we compose the word for the fibers and the word for the arrows. In the case of Figure 8 we have , , and .

From now on we assume is orientable. For the non-orientable case, see Theorem 3.2.

It is useful to label also the relations of the group presentation. As usual, denote the Wirtinger relations, that is a relation for every crossing as shown in Figure 10.

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Figure 10: Wirtinger relations.

We introduce the inner automorphism (conjugation) of a group in order to simplify the group relations:

For example, the Wirtinger relations may be rewritten as and , respectively.

The surface relation is

For every there are two relations and , associated to the edges and of :

For every there is a relation associated to the fiber

For every , the relation associated to the arrow is

The relation for every may have four different forms according to the following cases. The endpoint belongs to the edge and

The endpoint belongs to the edge and

The endpoint belongs to the edge and

The endpoint belongs to the edge and

Finally, for every , the relation of surgery is

Theorem 3.1.

Given a link in a Seifert fibered space (assume that the base surface is orientable) with an arrow diagram satisfying the above condition, we get the following presentation for the group of the link:

Remark.

When the genus of is zero, by applying a finite sequence of moves, we can assume that the link has no boundary points. This produces a major simplification of the group presentation.

In the case is non-orientable, consider the same generators except for the -s that are missing in the boundary identification of . The relations are modified as follows.

As usual, denote the Wirtinger relations. The surface relation becomes

For every , the edge relation is

For every , there is a fiber relation

For every , the relation associated to the arrow is

The relation for every depends on the such that the endpoint belongs to the edge ; the relation may have two different forms, according to . If the relation is

Otherwise

Finally, for every , the relation of surgery is

Theorem 3.2.

Given a link in a Seifert fibered space (assume that the base surface is non-orientable) with an arrow diagram satisfying the above condition, we get the following presentation for the group of the link:

We will prove only the orientable case, the proof of the unorientable case can be made in the same fashion.

Proof of Theorem 3.1.

Let be the fundamental polygon of the surface , base of the Seifert fibered space . Let be disjoint disks on , corresponding to the surgeries. Denote by the space . Consider the space , where is the equivalence relation that identifies the points of to the corresponding points of , moreover the relation identifies the points of the lateral surface , according to the boundary labels or and the signs or . Let be the quotient map. The Seifert fibered space is the result of the suitable -fillings on .

Consider the link . Up to small isotopies we can assume that is all contained inside , and can be represented as a system of arcs inside , that may end on the boundary.

The first goal is to compute , where is the basepoint on fixed as in Figure 9. In order to get a presentation of this group by the Seifert-Van Kampen theorem, we split into two parts. The first part is the tabular neighbourhood of and the second part is the “internal” part . Note that the first part deformation retracts to . The intersection between the two parts deformation retracts to .

As in the Wirtinger theorem for knots in , the fundamental group of can be presented by the generators associated to the overpasses, by the generators corresponding to the holes for surgeries and by the Wirtinger relations:

The space can be described by the following CW-complex:

0-complex

the base point ;

1-complexes

the loops of the surface , the loop of the fibration , the loops for the surgery holes, the loops corresponding to the arrows on the diagram, that means we have holes in created by , the loops on the lateral surface corresponding to the holes created by ;

2-complexes

there is one 2-complex that represents the surface and other 2-complexes corresponding to the surfaces that are parts of the lateral surface .

As a consequence, since the maximal tree is trivial, each -complex is a generator and each -complex is a relation.

The intersecting surface is a sphere with holes, hence its fundamental group is free and we label differently the generators according to the hole type: (resp. ) are the surgery holes in (resp. ), (resp. ) correspond to the arrow holes in (resp. ) and correspond to the lateral surface holes, indexed according to the corresponding overpasses indexation. As a consequence, is generated by .

By applying the Seifert-Van Kampen theorem we get the presentation:

After the deletion of the generators , the presentation becomes: