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Volume 2013 (2013), Article ID 549198, 9 pages
Cyclic Branched Coverings Over Some Classes of (1,1)-Knots
Dipartimento di Matematica, Università di Modena e Reggio E., Via Campi 213/B, 41100 Modena, Italy
Received 19 October 2012; Revised 12 March 2013; Accepted 18 March 2013
Academic Editor: Michel Planat
Copyright © 2013 Agnese Ilaria Telloni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We construct a 4-parametric family of combinatorial closed 3-manifolds, obtained by glueing together in pairs the boundary faces of polyhedral 3-balls. Then, we obtain geometric presentations of the fundamental groups of these manifolds and determine the corresponding split extension groups. Finally, we prove that the considered manifolds are cyclic coverings of the 3-sphere branched over well-specified -knots, including torus knots and Montesinos knots.
In this paper, we study the topological and covering properties of a class of closed connected orientable 3-manifold , depending on four nonnegative integer parameters such that , , , and . These manifolds are constructed from triangulated 3-cells, whose boundary faces are identified together in pairs. Then, we obtain finite -generator and -relator presentations for the fundamental group of the manifold which correspond to spines of the manifold. Furthermore, we determine the split extension of under the action of a cyclic group of order . It turns out to be the fundamental group of a three-dimensional closed orbifold obtained from the manifold as a quotient space with respect to the action of a suitable rotational symmetry of order . Finally, we completely describe the underlying topological space and the singular set of this orbifold. One of our results is that the combinatorial manifold is topologically homeomorphic to the -fold cyclic covering of the 3-sphere branched over a well-specified -knot. This allows us to draw explicitly the planar projection of a new interesting class of -knots (not yet considered in the literature) for which we study some geometrical and algebraic properties. As a very special case, that is, for certain values of the parameters, we obtain some torus knots or Montesinos knots as branching sets. Some subfamilies of our class of manifolds are known: the manifold , with , that is, (mod ), is homeomorphic to the closed connected orientable 3-manifold considered in ; furthermore, the triangulated 3-cells, from which the manifolds arise, are those used in  to construct a family of manifolds with totally geodesic boundary; finally, the manifold , with , that is, (mod ), is the unique closed 3-manifolds related with the class of hyperbolic 3-manifolds with totally geodesic boundary, combinatorially constructed in . We also show that our manifolds are contained in the Dunwoody family in the sense of , which was defined via suitable Heegaard diagrams. As a consequence, we give a simple polyhedral description of a large subclass of Dunwoody manifolds, and this connects with the general result obtained in . Finally, we observe that -knots are very important on the study of Dehn surgery. In fact, they are related with the unsolved problem of which knots in the 3-sphere admit Dehn surgery yielding lens spaces. In fact, in a recent paper , we have proved that our class of -knots admit Dehn surgeries yielding two infinite series of lens spaces.
2. The Combinatorial Manifolds
For every 4-tuples of nonnegative integers such that , , , and , let us consider the polyhedron , which is a triangulated 3-ball, having exactly boundary faces on each hemisphere (see Figure 1). Let us denote by (resp., ), , the southern (resp., northern) boundary faces of . The symbols 0 and in the picture denote the South pole and the North pole, respectively. The boundary face has oriented edges of type (denoted by ), oriented edges of type (denoted by ), and oriented edges of type (denoted by ). The vertices of are ordered in the clockwise manner according to the sequence , 0, , . The boundary face has oriented edges of type (denoted by ), oriented edges of type (denoted by ), and oriented edges of type (denoted by ). The vertices of are ordered in the counter-clockwise manner according to the sequence , , , , , , , . By convention, the polyhedron has no -edges, that is, every -vertex coincides with the -vertex labelled by the same subscript (see Figure 1).
Now, for every , we consider the homeomorphisms which identifies the oriented southern face as follows: with the oriented northern face by matching the oriented edges according to the above sequences (here the subscripts are taken modulo ). Obviously, this side pairing induces identifications among the vertices of the boundary faces (see Figure 1). The quotient space obtained from the polyhedron with the above-described side pairing is denoted by . A celebrated result of Seifert and Threlfall  states that the quotient space represents a closed connected orientable 3-manifolds if and only if its Euler characteristic vanishes. Using this criterion, we can prove our first result.
Theorem 1. For nonnegative integers , , and , the above constructed space is a closed connected orientable 3-manifolds if and only if ().
Proof. It is a routine matter to check that the quotient space has one vertex. So admits a cellular decomposition with one vertex, faces, one 3-ball, and edges. In fact, from the pairings of edges induced by the above-defined identification on the boundary faces of , for every , we get the following cycle of equivalent edges: where with , and is the minimum integer such that . Then, if (resp., ), we have (resp., ). The above sequence gives a closed edge path if and only if (mod ), that is, (mod ). Noting that , we get the congruence of the statement.
Let us assume the arithmetic conditions of Theorem 1 and denote by the fundamental group of the closed manifold . As before, set , with , and let be the minimum integer such that . Recall that if is a multiple of , then ; otherwise, if , then . By Theorem 1, we obtain immediately a geometric presentation for the group with the above-defined homeomorphisms , , as generators, and relations arising from the sequences of equivalent edges. In the following statements, we give such a presentation for the group and its dual obtained as edge-path group, since has exactly one vertex.
Corollary 2. Under the arithmetic conditions of Theorem 1, the fundamental group of the manifold admits a finite presentation with generators , , and cyclically defined relations of the form for every . Moreover, this presentation is geometric, that is, it corresponds to a spine of the combinatorial manifold .
Since the quotient complex has exactly one vertex (if the parameters satisfy the conditions of Theorem 1), we can obtain a further presentation for the fundamental group whose generators bijectively correspond to the loops of the 1-skeleton, and whose relations arise by walking around the boundaries of the 2-cells.
Corollary 3. Under the arithmetic conditions of Theorem 1, the fundamental group of the manifold admits a further presentation with generators and cyclic relations for every . This presentation is also geometric, that is, it corresponds to a spine of the manifold .
The Heegaard diagrams coming from the polyhedral description of the manifolds are exactly the ones defined by Dunwoody in 1995 (see ). Then, are just particular cases of the so-called Dunwoody manifolds , which depend on six integer parameters. Such manifolds have been extensively studied by many researchers (see, e.g., [4, 8–13]). More in detail, we have the following.
Corollary 4. Under the arithmetic conditions of Theorem 1, the above-constructed closed connected 3-manifold is precisely the Dunwoody manifold .
From this result, we obtain immediately a simple polyhedral description of a large subclass of Dunwoody manifolds, which connects with the general result given in .
3. Split Extension Group
The above-constructed polyhedron admits a rotational symmetry of order , denoted by , with respect to its central axis, that is, the axis connecting the points and . The automorphism maps every boundary face in the next face , where the subscripts are considered modulo . This automorphism induces a symmetry, also denoted by , on the combinatorial manifold . Let us denote by the split extension group of with respect to the cyclic group of order generated by . From the obtained presentations for the group , we get two different presentations for the split extension group .
Theorem 5. For nonnegative integers ,, , , and (), the split extension group of admits the finite presentations where , , and is the minimum integer such that .
Proof. Setting , where , in the cyclic relations of as written in the statement of Corollary 2, we get Let be with inverse relation . Then, the word of the previous relation becomes which is equivalent to Substituting and (i.e., ) gives the presentation Now, we determine the second presentation for the group . Substituting , where , in the cyclic relations of (as written in the statement of Corollary 3) yields where is the maximum between 0 and . If (and hence , then the previous relation is equivalent to Setting and , with inverse relations and , we obtain the presentation listed in the statement. Analogously, if , hence , then the previous relation becomes As done before, setting and , we obtain the requested relation.
4. Topological and Covering Properties
The split extension group turns out to be the fundamental group of the 3-dimensional closed orbifold obtained as quotient space of the manifold under the action of the rotational symmetry of order . Recall that this symmetry is induced by the rotation of order , also denoted by , with respect to the interior axis of the polyhedron which connects the points 0 and . The above orbifold will be denoted by , where , , ; in fact, it does not depend by the parameter , which represents the shift of the northern boundary faces of the polyhedron with respect to the southern ones. We recall that a knot in a lens space is said to be a -knot if there exists a genus one Heegaard splitting , where is a solid torus, is a properly embedded trivial arc, for , and is an attaching homeomorphism. An arc properly embedded in a solid torus is said to be trivial if there is a disk in with and (see, e.g., ). Set . The pair is also called a -splitting of (.
Theorem 6. For nonnegative integers , , , , and (), the combinatorial closed 3-manifold is homeomorphic to the -fold cyclic covering of the 3-sphere branched over the -knot depicted in Figure 6, where and .
Proof. The underlying space of the orbifold is just the manifold . This manifold can be represented by the Heegaard diagram of genus one depicted in Figure 2. Since the fundamental group is trivial, . By Theorem 6 of , the singular set of the orbifold is a -knot , depending on three parameters and formed by the image of the 0-axis of (depicted as a dotted line in Figure 2) in the quotient space . The Heegaard diagram of in Figure 2 can be transformed into a simpler one via suitable Whitehead-Zieschang reductions (see ).
Each of these moves consists in a simplification of the graph along a closed simple curve surrounding one of the circles in a planar representation of the diagram (for more details about such moves see, e.g., [15–17]). This allows us to obtain a Heegaard diagram having a reducible 1-handle; this diagram admits different planar representations depending on being even or odd. In Figure 3(a), we show a Heegaard diagram of with a reducible 1-handle for odd.
In order to cancel the unique 1-handle, we can draw the singular set (depicted as a dotted line in Figures 2 and 3) into a cylinder, so that the circles and coincide with its bases (see Figure 3(b), for odd). Then, we image to unscrew the upper base of the cylinder times and the lower one times. This gives a braid with threads on its upper side and threads on its lower side. Now it only remains to close the braid by identifying the 0- and -poles of the singular set. The obtained knot , shown in Figure 4, has so many twists in its left (resp., right) side as the unscrew motions on the upper (resp., lower) base of the cylinder. An equivalent planar projection of the knot , which is the singular set, is given in Figure 5. By a sequence of Reidemeister moves, we can further simplify the knot into the cyclic form shown in Figure 6. The case even can be treated in a similar way. Moreover, the final planar projection depicted in Figure 6 represents the knot for both cases odd and even.
Theorem 7. For nonnegative integers , , , , and (), the group of the -knot admits the 2-generator presentations where , and is the minimum integer such that .
Proof. The group can be embedded as a normal subgroup of index in , whose presentation is given in Theorem 5. The map , defined by , gives the desired embedding. Furthermore, is isomorphic to the normal closure of in , and there is a short exact sequence , where . Let us consider the 3-orbifold in the proof of Theorem 6, whose underlying space is and whose singular set is the -knot , with branching index . The -fold covering map induces a group embedding , where denotes the fundamental group of . In particular, we have , and fits in a short exact sequence , where is generated by the rotational symmetry . Now, five lemma implies , and is presented as above by Theorem 5. In fact, a -knot is a two-generator knot (see [9, 14]). Therefore, a -knot in the 3-sphere is prime (see ). But prime knots are classified by their groups.
To end this section, we consider the manifolds for particular values of the parameters and explicitly recognize the corresponding branch sets among some classical knots.
Theorem 8. For nonnegative integers , , , , , and (), the combinatorial manifold is the -fold cyclic covering of the 3-sphere branched over the torus knot , where and .
Proof. Under the considered hypotheses, the split extension group admits the finite presentation (see the second presentation in the statement of Theorem 5). This is the fundamental group of the orbifold , having the -knot as singular set. Since a -knot is a two-generator knot (see, e.g., ), it is a prime knot in the 3-sphere . Thus, is completely determined by its group (for more details see, e.g., [19, Theorem 6.1.12, page 76]). In this case, the group of is presented by , where is a meridian-longitude pair of the knot. Setting , with inverse relation , the group of is presented by . Since the transformation matrix between the pairs and is with determinant +1, we see that also is a meridian-longitude pair of the knot. The last group presentation just encodes the torus knot of type , with and .
Theorem 9. For nonnegative integers , , , , and (), the combinatorial manifold is the -fold cyclic covering of the 3-sphere branched over the torus knot , where and .
Proof. In this case, the -knot is uniquely determined by its group . Substituting , with inverse relation , yields . Hence, is just the torus knot of the statement.
Theorem 10. For every , the -knot , where and , depicted in Figure 7(b), is encoded by the 2-generator group with defining presentations If further , then the knot is equivalent to the Montesinos knot (in particular, the torus knot for ) encoded by the 2-generator group with defining presentations . Furthermore, the combinatorial manifold is the Seifert fibered space with Seifert invariants .
Proof. The first sentence follows from Theorem 7 and Theorem 3.7 of . If and , then the singular set is the knot in Figure 7(a). It is equivalent to the Montesinos knot , which is just the Montesinos knot of the statement (for more information on Montesinos knots and links see , Chapter 12, Section D). The last statement follows from a well-known theorem of Montesinos (see, e.g., , Proposition 12.30).
This work is supported by the project “L’Orèal Italia Per Le Donne e La Scienza.” The author thanks A. Cavicchioli (University of Modena and Reggio E., Italy) and L. Paoluzzi (University of Marseille, France) for their comments and many helpful suggestions.
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