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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 103209, 7 pages
The Dual and Mirror Images of the Dunwoody 3-Manifolds
Department of Mathematics, Dong-eui University, Busan 614-714, Republic of Korea
Received 1 April 2013; Revised 1 July 2013; Accepted 22 July 2013
Academic Editor: Seppo Hassi
Copyright © 2013 Soo Hwan Kim and Yangkok Kim. 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.
Recently, in 2013, we proved that certain presentations present the Dunwoody -manifold groups. Since the Dunwoody -manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody -manifolds. It is well known that every -knots in a lens space can be represented by the set of the 4-tuples (Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody -manifolds, we proved the fact that the certain subset of representing all -bridge knots of -knots is determined completely by using the dual and mirror -decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).
In the study of the Dunwoody -manifolds, Heegaard splittings and Heegaard diagrams provide a simple means of understanding the Dunwoody -manifolds by changing a -dimensional problem into a -dimensional problem, for the basic definitions of Heegaard splittings and Heegaard diagrams; see [1, 2]. Even with the help of Heegaard diagrams, because the Dunwoody -manifolds do not have a unique Heegaard diagram, generally, the following are still open questions about -manifolds: given -manifolds and , (homeomorphism problem) is ? or (isomorphism problem) is ? The Dunwoody -manifolds have a Heegaard diagram, from which one can obtain presentations for a group; however, not all group presentations arise from Heegaard diagrams of Dunwoody -manifolds. The Heegaard diagrams are useful for understanding properties of the Dunwoody -manifolds as there is a correspondence between the Heegaard diagrams and the fundamental groups for the Dunwoody -manifolds, allowing transformations of a group presentation for the fundamental groups to a simple calculus of the Heegaard diagrams. Thus, transformations of the Heegaard diagrams are corresponding to transformations of a group presentation for the fundamental groups of the Dunwoody -manifolds. In , we gave some conditions for the answer to the generalized Dunwoody -manifolds, constructed by group presentations for the fundamental group. For an answer to the above problems between the Dunwoody -manifolds, we consider the dual and mirror images of the Dunwoody -manifolds and give the basic definitions about them in the following.
Let be a Heegaard splitting of a -manifold with genus . A properly embedded disc in the handlebody is called a meridian disc of if cutting along yields a handlebody of genus . A collection of mutually disjoint meridian discs in is called a complete system of meridian discs of if cutting along gives a -ball. Let denote the -sphere which lies in the closed orientable surface of genus . The system is said to be a Heegaard diagram of the -manifold and denoted by . On the other hand, the system is called a dual Heegaard diagram of the -manifold if is a complete system of mutually disjoint meridian discs in , and is the -sphere which lies in the closed orientable surface of genus . In other words, is the dual Heegaard diagram of .
For , we call and each -knot and lens space, and in this paper, all lens spaces are assumed to include , and . Then, admits a -decomposition provided that there exists a Heegaard splitting of genus one of such that is a Heegaard diagram of and and are properly embedded trivial arcs, where is an attaching homeomorphism. By the dual -decomposition of we mean that is the dual Heegaard diagram of instead of .
In 1995, Dunwoody introduced the -tuples yielding a family of genus Heegaard diagrams of closed orientable -manifolds called the Dunwoody -manifolds . Moreover, the Dunwoody -manifolds are determined by the -fold strongly cyclic coverings of lens spaces branched over -knots , defined by the monodromy , where is the cyclic group of order , [3, 5–8]. In fact, such branched sets in the quotient spaces of the Dunwoody -manifolds by a cyclic action of order are representing the -knots in lens spaces (possibly the -sphere) [7, 9], and some classes of such knots represented by the Dunwoody -manifolds contain all -knots in .
Now, in view of -knot , the Dunwoody -decomposition of determined by two permutations and and -tuples of integers as follows was introduced in [3, 8, 10] (see Figure 1), where each cycle of corresponds to the end points of line segments in the Heegaard diagram as in Figure 1, and each cycle of corresponds to a pair of end points which is identified in forming the handlebody . For each , we denote the Dunwoody -decomposition of by and the Dunwoody -knot , represented from , by . Note that for each point on , the simple closed curve on is determined by the repeated applications of and as follows: which forms exactly an orbit of . It is well known that all -knots in are to be the Dunwoody -knots, but the representation of -knots by -tuples is not unique. For example, and represent the pretzel knot as a -knot. However all types of the Dunwoody -decompositions representing all -bridge knots were determined completely in [7, 10], and moreover, the types of the Dunwoody -decompositions representing the certain class of torus knots are given in [6, 11, 12]. For each , the -tuples satisfying conditions and induce the Dunwoody -manifolds, denoted by , as closed orientable -manifolds, where , , and are some integers defined in [4, 8]. Thus, the Dunwoody -manifold is the -fold strongly cyclic covering space of a lens space branched over the Dunwoody knot .
The Dunwoody -manifolds play an important role in determining which cyclically presented group corresponds to a -manifold; see [13, 14]. Indeed, in order to study a -manifold with some particular group as the fundamental group, the Dunwoody -manifolds have a Heegaard diagram from which one can obtain a presentation for the group. Thus, to find the Dunwoody -manifolds, one must seek a cyclically presented group associated with a -manifold. Furthermore, as in , the branched covering spaces of the spatial -curves containing -knots as the constituent knots are related to the Dunwoody -manifolds. Therefore, the concept of the Dunwoody -manifolds is important in knots, branched coverings, and graph theories. Moreover, recent manuscripts of the Dunwoody -manifolds can be found in [3, 8, 10, 12, 16–19].
Moreover, the -decompositions obtained from the Dunwoody -manifolds are very useful for transformations of the Heegaard diagrams. In this paper, we denote the dual decomposition of by or , which has the Heegaard diagram of , and the mirror decomposition of by or . The Dunwoody -manifolds obtained from the dual and mirror decompositions of are called the dual and mirror images of the Dunwoody -manifold , respectively, where . For a specific example, see .
In Section 2, we show that the defined regions in the Dunwoody -decompositions are divided by inessential and essential subregions. Moreover, we show that the regions in the Dunwoody -decompositions representing all lens spaces containing are connected. In Section 3, we introduce an algorithm to obtain from by using matrices related to the certain sequences of . Moreover, we introduce an algorithm to obtain the mirror -decomposition as the mirror image of under an orientation-reversing autohomeomorphism by using matrices. Thus, our results generalize some results in [7, 10], classifing the types of the Dunwoody -decompositions representing -bridge knots.
2. On the Regions of
We assume that is the simple closed curve of , and that is the region of . Then, is a disconnected region if is the inessential loop in the surface , and otherwise, is a connected region. Therefore, there are three kinds of loops on the : is the inessential loop, is the meridian loop, and is the loop that is isotopic to torus knot. In case of , is denoted by ; of , is homeomorphic to and denoted ; and of , it is homeomorphic to the lens space or the -sphere and denoted by , where is the integer defined in . In fact, is homeomorphic to the -sphere if and only if is connected and .
Let be the Dunwoody -decomposition of . Since , the number of simple closed curves on is one, denoted by , and thus, is to be a lens space . Since and have the same parity, the Dunwoody -decomposition with even is representing the lens space containing (for ), but it is not representing -sphere even if is connected; see Corollary 3.6 in . For example, the region of with and is disconnected and both and with have a connected region and represent .
Lemma 1. If the region is disconnected, both and of have even numbers.
Proof. The Dunwoody -decomposition with disconnected is obtained from by isotopic moves of inessential loop on . Thus, and are the result of an increase by an even number from and of . Thus, both and of have even numbers.
Recall that each cycle of corresponds to the end points of line segments in the Heegaard diagram as in Figure 1 and each cycle of corresponds to a pair of end points which is identified with and . In Figure 1, we consider a cyclic sequence on with starting point and a cyclic sequence of points with starting point that and cross by repeated applications of and as follows:
A subregion of is said to be inessential in if the subregion is bounded by of and of as Figure 2, and otherwise, the subregion is said to be essential in .
Theorem 2. Let be the region of . Then, has the inessential subregion if and only if is disconnected.
Proof. Suppose that the region of has the inessential subregion. Then, the simple closed curve is isotopic to an inessential loop on . By Lemma 1, is disconnected. Now assume that the region of is disconnected. By Lemma 1, and are even numbers which are obtained by increasing as even numbers from and of . Thus, has the inessential subregion.
Corollary 3. Let be the region of . Then, has the essential subregion if and only if is connected.
Corollary 4. Let and be the integers defined in  and the region of . If is a disconnected region, then both and are equal to zero.
For counterexample, the region of has and , but it has no inessential subregion; that is, it is the connected region.
Therefore, we have the partition of the Dunwoody -decompositions as the following diagram in Figure 3. In Figure 3, is a set of the Dunwoody -decompositions having a disconnected region, is a set of the Dunwoody -decompositions satisfying and , and is a set of the Dunwoody -decompositions having a connected region. Note that , , and are examples for , , and , respectively.
In the remainder of this paper, we work in the connected region unless otherwise specified. In the connected region , we assume two cyclic sequences and as follows:
3. The Dual and Mirror Images of
We introduce an algorithm to obtain from as follows. Let be the Dunwoody -decomposition of . Suppose that is the -decomposition in view of as Figure 1 and that is the dual decomposition in view of . Let and be the meridian disk and the oriented simple closed curve on . Then, it is understood easily from the attaching homeomorphism that the images of and on are defined as the meridian curve and the oriented simple closed curve on , respectively.
We consider a cyclic sequence on with starting point and a cyclic sequence of points with starting point and cross by repeated applications of two permutations and defined in [3, 8, 10] as . Let and be sets of vertices in and (see Figure 1), respectively. For each , let be a consecutive -series in and a consequence -series in . Then, has two matrices such that if is the beginning point of a curve corresponding to (resp., ); then, we define (resp., ) as -entries in matrix and (resp., ) as -entries, and otherwise (i.e., is the final point), we define (resp., ) as -entries and (resp., ) as -entries; see Figure 4 for each of and . In fact, for each point , in case of , two permutations and induce matrices denoted by and in case of , two permutations and induce In particular, has matrix such that and are -entries and -entries, respectively, and has matrix such that and are -entries and -entries, respectively. The matrices are denoted by Therefore, we have matrix corresponding to a cyclic sequence along as follows:
Considering a sequence and , has matrix such that is -entries, and has matrix such that is -entries. Therefore, we have matrix corresponding to for a cyclic sequence as follows:
Theorem 5. Let be the Dunwoody -decomposition and . Then, there exist two matrices and such that(1)the number of submatrices and in corresponding to consecutive 2-series of is equal to , (2)let be a point of meeting and consecutive 3-series in ; then, if , and otherwise, , (3).
Proof. Let and be matrices corresponding to and along and , respectively. Then, submatrices and of are representing curves of meeting and . Thus, the number of such submatrices of is .
Given and , the point in is corresponding to the point in by . If the corresponding submatrix is , then and is the beginning point of . If the corresponding submatrix is , then , and is the terminal point of . These are determined by an orientation of the curve . For each of the cases, is the point in , so means the number of curves of meeting .
It is proved from the fact that .
Let and be the meridian disk and the oriented simple closed curve on corresponding to , respectively. Then, and are the meridian disk and the orientable closed curve on corresponding to , respectively, where is an attaching homeomorphism for the -decomposition . We obtain from by the following facts and the properties of Theorem 5.
By the definition of the dual process, the role of will be changed into on . Thus, we obtain two cyclic sequences and from and , respectively, for as in the following:
Moreover, two matrices and for and can be obtained from and , respectively. If in , in the bigon bounded by , is a starting point in , then in will be starting point in . Moreover, the number is equal to the number situated on after converting on the sequence to . Thus, we can rearrange terms of and as follows.(1)We replace each term of with terms of an arithmetic sequence with first item , common difference , and final term and denote it by . (2)Furthermore, we rearrange terms of , , and by new terms along the rule of and denote each of them by , and . (3)th term of will be the number of .
The following corollary follows from the properties of Theorem 5.
Corollary 6. Let be the Dunwoody -decomposition and . Then, the dual -decomposition of and is obtained by the following: (1)the number of submatrices or in corresponding to consecutive 2-series of is equal to ; (2)let and consecutive 3-series in . Then, if and otherwise ; (3). On the contrary, one can obtain from by the dual process of the above.
We denote by the mirror image of under an orientation-reversing autohomeomorphism , where we orient with orientation induced from by . Then, the orientations of , , and may be changed the other way in , and so the role of in will be changed into in . Considering a cyclic sequence on with starting point and a cyclic sequence of points with starting point ; then, and cross by repeated applications of and of as follows By definition of , it is equivalent to Putting as a starting point in , we can easily find such that .
Corollary 7. Let be the Dunwoody -decomposition and . Then, the mirror image of is , which is obtained from the corresponding matrices as follows: where , , and .
Remark that another proof which represents the mirror image was obtained in . Here, we give the canonical examples in the following.
Example 8. Let be the Dunwoody -decomposition, in fact, it is representing the torus knot in . Then, the following are cyclic sequences corresponding to and , respectively, for : If we replace and with and , respectively, then are cyclic sequences for . Replacing by as a starting point in , we can rearrange and as follows: The corresponding matrices of them are Since has the four submatrices as follows: we have by Corollary 6. The following is a result of Corollary 6. Since of corresponds to of ; thus, . From and , we have . For in , we have , so . Thus, , and we have .
Example 9. To obtain the mirror image of , we consider two cyclic sequences which are images of of cyclic sequences and for crossing of and , where each number in cyclic sequences is equal to . Then, the corresponding matrices of and are If we replace with in order to rearrange follows then is to be , and so ; that is, . Using , , and of Corollary 6 for and , we can determine that , , and ; that is, .
Remark 10. As the same problem for , we have and .
Summarizing, we have the following.
Corollary 11. For any -knot in the lens space, the Dunwoody -decompositions for commute as follows:
Moreover, the Dunwoody -manifolds obtained from the Dunwoody -decompositions commute, for some , where .
Lastly, we consider two presentations to be equivalent if they differ by an isomorphic. Then, the isomorphism problem is hard because the map from the presentation to the Heegaard diagram is not one-to-one. So the isomorphism problem is obviously not solvable, but we can determine that at least four presentations in Corollary 11 are equivalent; that is, they are group presentations of the fundamental group for a closed orientable -manifold. Moreover, results in this paper show that there are at least four Dunwoody -decompositions representing a given -knot. Table 1 shows the Dunwoody -decompositions and their dual decompositions of -knots in by , and , , , and means a 2-bridge knot, a torus knot, a pretzel knot, and knots classified by Rolfsen table, respectively. The last column in Table 1 represents a -bridge knot induced by the standard involution on for some .
This work was supported by Dong-eui University, Grant no. 2012AA085.
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