HEEGAARD EXAMPLES NOTE: This file contains several examples which should be helpful in getting acquainted with the program. EXAMPLE 1: A PRESENTATION OF L(5,2) _______________________________________________________________________________________________ Here is a simple 2-generator, 2-relator presentation which is uniquely realizable. R 1: AAB R 2: Abb Try working this example by hand, or start the program and run Example 1. If you have the program run this example, the program should produce output like the following: _______________________________________________________________________________________________ The initial presentation was: R 1: AAB R 2: Abb This presentation has length 6 and is freely reduced. The data appears consistent. The presentation is realizable. This manifold is closed. <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 2 Generator(s) Length 6 From Presentation 1 IP presents the Lens space L(5,2). R 1) AAB R 2) Abb The program performed 0 automorphism(s), formed 0 band sum(s) and examined 2 diagram(s). Left to do 3. _______________________________________________________________________________________________ The program claims the initial presentation yields a 3-manifold homeomorphic to L(5,2). _______________________________________________________________________________________________ EXAMPLE 2 A PRESENTATION OF L(5,1) _______________________________________________________________________________________________ Here is another 2-generator, 2-relator presentation which is also a fairly simple and which is uniquely realizable. R 1: AAABB R 2: AABBB Try working this example by hand, or start the program and run Example 2. If you have the program run this example, the program should produce output like the following: _______________________________________________________________________________________________ The initial presentation was: R 1: AAABB R 2: AABBB This presentation has length 10 and is freely reduced. The data appears consistent. The presentation is realizable. This manifold is closed. <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 2 Generator(s) Length 10 From Presentation 1 IP presents the Lens space L(5,1). R 1) AAABB R 2) AABBB The program performed 1 automorphism(s), formed 0 band sum(s) and examined 2 diagram(s). Left to do 3. _______________________________________________________________________________________________ The program claims this presentation yields a 3-manifold homeomorphic to L(5,1). _______________________________________________________________________________________________ EXAMPLE 3: REMOVING A PAIR OF SEPARATING VERTICES VIA LEVEL-TRANSFORMATIONS _______________________________________________________________________________________________ Here is a simple example which illustrates how level-transformations can sometimes be used to transform a presentation whose reduced Whitehead graph has a pair of separating vertices into a presentation whose reduced Whitehead graph has no pair of separating vertices. R 1: AABCaBacAcbabC You can have the program run this as Example 3. If you run this example, the program should produce output which looks like the following: _______________________________________________________________________________________________ The initial presentation was: R 1: AABCaBacAcbabC This presentation has length 14 and is freely reduced. The Whitehead graph has a pair of separating vertices. The data appears consistent. After some level transformations: The presentation is realizable. This relator separates the Heegaard surface. <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 3 Generator(s) Length 14 From Presentation 1 IP vertex 'A' and vertex 'a' separate the diagram. R 1) AABCaBacAcbabC Presentation 2 of Component 1: 3 Generator(s) Length 14 From Presentation 1 Lt via level transformations of sep_pairs. R 1) AAABCBaaccbabC The program performed 1 automorphism(s), examined 0 bandsum(s) and examined 2 diagram(s). Left to do 0. _______________________________________________________________________________________________ 1) Observe that the pair of vertices {A,a} separate the reduced Whitehead graph of the oroginal presentation. 2) Note that applying the automorphism C -> CA yields: R 2: AAABCBaaccbabC 3) Note that R2 again has length 14, so the automorphism is level. 4) Note that the reduced Whitehead graph of R2 has no pair of separating vertices. _______________________________________________________________________________________________ EXAMPLE 4: A PRESENTATION WHOSE REALIZATION IS NOT UNIQUE BECAUSE A VALENCE-TWO ANNULUS EXISTS. _______________________________________________________________________________________________ Here is a fairly simple example of a presentation which is realizable but whose realization is not unique because a Valence-Two Annulus exists. R 1: AAABAAB R 2: BCbc R 3: CDcd R 4: BD You can have the program run this as Example 4. If you run this example, the program should produce output which looks like the following: _______________________________________________________________________________________________ The initial presentation was: R 1: AAABAAB R 2: BCbc R 3: CDcd R 4: BD This presentation has length 17 and is freely reduced. The data appears consistent. The presentation is realizable. However, the realization is not unique because an annulus exists. These relators separate the Heegaard surface. <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 4 Generator(s) Length 17 From Presentation 1 IP There exists an annulus which swallows vertice(s) {a,A} and otherwise follows the curve: B R 1) AAABAAB R 2) BCbc R 3) CDcd R 4) BD The program performed 0 automorphism(s), examined 0 bandsum(s) and examined 1 diagram(s). Left to do 0. _______________________________________________________________________________________________ 1) Observe that the presentation has a reduced Whitehead graph which has a unique embedding in the two-sphere S2. And note that the Whitehead graph of this presentation also has a unique embedding in S2. 2) Observe that both vertex 'A' and vertex 'a' have valence two in the reduced Whitehead graph of the presentation. 3) Let D be a Heegaard diagram which realizes this presentation. Observe that there is an annulus N in D such that the boundary components of N are disjoint from the attaching curves of the 2-handles representing R1 -> R4, and N "swallows" vertice(s) {a,A} and otherwise follows the curve: B. 4) Observe that the Heegaard diagram is not unique because N can be reembedded in D by applying a homeomorphism which is a reflection of N in its centerline and which exchanges the boundary components of N. _______________________________________________________________________________________________ EXAMPLE 5: A PRESENTATION WITH A MORE GENERAL ANNULUS. _______________________________________________________________________________________________ Here is an example which, like the previous example, contains an annulus. However, this example is more pathological, since the present version of the program is unable to determine whether this presentation is realizable. R 1: AAAABBBCCCCDDD R 2: AAAABBB R 3: AAABB R 4: CCCDD (You can have the program run this as Example 5.) This is actually a realizable presentation of the 3-sphere S3. However, because the annulus exists, the program can't find a realization, and so the program reports that it is unable to determine whether this presentation is realizable. If you have the program run this example, it should produce output like the following: _______________________________________________________________________________________________ The initial presentation was: R 1: AAAABBBCCCCDDD R 2: AAAABBB R 3: AAABB R 4: CCCDD This presentation has length 31 and is freely reduced. The Whitehead graph has a pair of separating vertices. Unable to remove the separating pair of vertices. Trying to reduce the genus of the presentation. Unable to determine whether the presentation is realizable. <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 4 Generator(s) Length 31 From Presentation 1 IP Vertices 'A' and 'b' separate the diagram. The component consisting of vertice(s) {C,c,D,d} lies in an annulus which swallows the component and otherwise follows the curve: AAAABBB R 1) AAAABBBCCCCDDD R 2) AAAABBB R 3) AAABB R 4) CCCDD The program performed 0 automorphism(s), examined 0 bandsum(s) and examined 1 diagram(s). Left to do 8. _______________________________________________________________________________________________ Observe that unlike the Valence-Two Annulus case, the Whitehead graph of this presentation has a pair of separating vertices. (Indeed, every presentation which can be obtained from this presentation via level-transformations, will also have a pair of separating vertices. So level-transformations are of no help in this situation.) _______________________________________________________________________________________________ EXAMPLE 6: 18 SURGERY ON THE (-2,3,7) PRETZEL KNOT. _______________________________________________________________________________________________ This is a rather long example, but it illustrates many of the things the program does. Relators 1 -> 12 below are a Wirtinger presentation of the (-2,3,7) pretzel knot with one relator for each crossing. The program's routine Wirtinger() recognized that this presentation was probably the presentation of a knot, with meridian M = A, and longitude L = IJKGHAEABCDGaaaaaaaaaaaa, and offered the opportunity to perform Dehn surgery on the knot. I choose to perform 18 surgery, so Wirtinger() appended Relator 13, which represents the additional surgery relation, to the presentation. _______________________________________________________________________________________________ The initial presentation was: R 1: LGag R 2: FAga R 3: DGeg R 4: GEhe R 5: EHfh R 6: HAia R 7: AIbi R 8: IBjb R 9: BJcj R 10: JCkc R 11: CKdk R 12: KDld R 13: AIJKGHAEABCDGaaaaaaaaaaaaAAAAAAAAAAAAAAAAA _______________________________________________________________________________________________ The program then freely reduced this presentation, and used its Canonical_Rewrite() routine to rewrite the presentation. Next, I elected to have the program reduce and simplify this presentation, and I opted to have it do so by performing bandsums and by eliminating all appearances of primitives among the relators. Then I put the program into its MicroPrint mode so that it would save the following log of its activities. _______________________________________________________________________________________________ This presentation has length 90 and freely reduces to length 66. The rewritten initial presentation is: R 1: AAAAAABCDEAFAGBHIJ R 2: ABKb R 3: AGaj R 4: ALaB R 5: AjeJ R 6: BCbf R 7: BfgF R 8: CHcK R 9: CHdh R 10: DIdh R 11: DIei R 12: EJei R 13: FgLG _______________________________________________________________________________________________ Next, the program performed the following automorphisms on the initial presentation to reduce it to minimal length, rewrote the resulting presentation, and saved it as Presentation 1. Then, the program discovered that the Whitehead graph of Presentation 1 had a pair of separating vertices, and so the program called its routine Level_Transformations() in an attempt to slide components of the separation around in order to arrive at a minimal length presentation without any pairs of separating vertices in its Whitehead graph. _______________________________________________________________________________________________ Do Aut 1 time(s): B->aB F->aFA G->aGA L->aLA Do Aut 1 time(s): C->bC K->bKB Rewrote the presentation using the substitution: AGBCDEFHIJKL Saved the current presentation as: Presentation 1 The Whitehead graph has a pair of separating vertices. Performed a level-transformation by sliding vertice(s): {K} along a path represented by: a to obtain the presentation: R 1: AAAAABCDEAFGHIJ R 2: AgeFE R 3: BHchg R 4: AjdJ R 5: BHbaK R 6: CIch R 7: CIdi R 8: DJdi R 9: EfLF R 10: Alg R 11: Bge R 12: K R 13: Fj Performed a level-transformation by sliding vertice(s): {l} along a path represented by: aG to obtain the presentation: R 1: AAAAABCDEAFGHIJ R 2: AgeFE R 3: BHchg R 4: AjdJ R 5: BHbaK R 6: CIch R 7: CIdi R 8: DJdi R 9: EfLgAF R 10: l R 11: Bge R 12: K R 13: Fj _______________________________________________________________________________________________ After performing two slides, the program found a presentation without any pairs of separating vertices in its Whitehead graph, and saved the resulting rewritten presentation as Presentation 2. Then the program verified that this presentation is realizable. _______________________________________________________________________________________________ Rewrote the presentation using the substitution: ABCDEFGHIJlK Saved the current presentation as: Presentation 2 The data appears consistent. After some level transformations: The presentation is realizable. These relators separate the Heegaard surface. Started with presentation 2, Length 59: R 1: AAAAABCDEAFGHIJ R 2: AFEfKg R 3: ABhbL R 4: AgeFE R 5: BHchg R 6: AjdJ R 7: CIch R 8: CIdi R 9: DJdi R 10: Bge R 11: Fj R 12: K R 13: L _______________________________________________________________________________________________ Next, the program starts looking for primitive relators which can be used to reduce the genus of M; looking first for defining relators of length 1 and 2. _______________________________________________________________________________________________ Relator 13 is a defining relator which was used to eliminate generator L. The presentation is currently: R 1: AAAAABCDEAFGHIJ R 2: AFEfKg R 3: ABhb R 4: AgeFE R 5: BHchg R 6: AjdJ R 7: CIch R 8: CIdi R 9: DJdi R 10: Bge R 11: Fj R 12: K Relator 12 is a defining relator which was used to eliminate generator K. The presentation is currently: R 1: AAAAABCDEAFGHIJ R 2: AFEfg R 3: ABhb R 4: AgeFE R 5: BHchg R 6: AjdJ R 7: CIch R 8: CIdi R 9: DJdi R 10: Bge R 11: Fj Relator 11 is a defining relator which was used to eliminate generator F. Rewrote the relators by replacing J with F. The presentation is currently: R 1: AAAAABCDEAFGHIF R 2: AFEfg R 3: ABhb R 4: AgeFE R 5: BHchg R 6: AfdF R 7: CIch R 8: CIdi R 9: DFdi R 10: Bge _______________________________________________________________________________________________ When there are no defining relators of length 1 or 2, the program searches the relators in random order looking for any other defining relators. If it finds a defining relator, the program uses that relator to eliminate a generator from the presentation. And, if the defining relator could be used to eliminate more than one generator, then the program chooses a generator, at random, from among the possibilities, and uses the defining relator to eliminate the chosen generator from the presentation. _______________________________________________________________________________________________ Relator 10 is a defining relator which was used to eliminate generator B. Rewrote the relators by replacing I with B. The presentation is currently: R 1: AAAAAEGCDEAFGHBF R 2: AFEfg R 3: AEGhge R 4: AgeFE R 5: EGHchg R 6: AfdF R 7: CBch R 8: CBdb R 9: DFdb _______________________________________________________________________________________________ After eliminating generator B, the program finds that the presentation no longer has minimal length; so it performs the following automorphisms to again reduce the presentation to minimal length. _______________________________________________________________________________________________ Do Aut 1 time(s): H->gH Do Aut 1 time(s): G->eG H->eH Do Aut 1 time(s): E->AEa F->Fa H->AH Do Aut 1 time(s): A->fAF E->fE G->fG H->fH 4 automorphism(s) reduced the length to 46. The presentation is currently: R 1: AAAAGCDfAEFeHB R 2: EfgAE R 3: Gh R 4: gFE R 5: HchEf R 6: Ad R 7: CBchaG R 8: CBdb R 9: DaFdb Rewrote the presentation using the substitution: AHCDFeBG Saved the current presentation as: Presentation 3 Started with presentation 3, Length 46: R 1: AAAABCDEAFefGH R 2: AGChcb R 3: AFFEb R 4: AdHDE R 5: CgefG R 6: CHdh R 7: BfE R 8: Ad R 9: Bg Relator 8 is a defining relator which was used to eliminate generator A. Swapped Relator 8 with Relator 9 and reduced the number of relators. Rewrote the relators by replacing H with A. Free reductions reduced the length of the current presentation from 44 to 42. The reduced presentation is: R 1: DDDDBCDEDFefGA R 2: DGCacb R 3: DFFEb R 4: ADE R 5: CgefG R 6: CAda R 7: BfE R 8: Bg Relator 8 is a defining relator which was used to eliminate generator G. The presentation is currently: R 1: DDDDBCDEDFefBA R 2: DBCacb R 3: DFFEb R 4: ADE R 5: CbefB R 6: CAda R 7: BfE Relator 2 is a defining relator which was used to eliminate generator D. Swapped Relator 2 with Relator 7 and reduced the number of relators. Rewrote the relators by replacing F with D. Free reductions reduced the length of the current presentation from 70 to 52. The reduced presentation is: R 1: BCAAAABCAcbEBCAcbDedBA R 2: BdE R 3: CAcbDDE R 4: ABCAcbE R 5: CbedB R 6: CABCacba Do Aut 1 time(s): B->Bc Do Aut 1 time(s): D->BDb E->BEb Do Aut 1 time(s): B->Ba C->Ca E->Ea 3 automorphism(s) reduced the length to 35. The presentation is currently: R 1: AAABEDAedAcAB R 2: cBadE R 3: DDEbC R 4: BEbA R 5: Ced R 6: BabaC Rewrote the presentation using the substitution: abCDE Saved the current presentation as: Presentation 4 Started with presentation 4, Length 35: R 1: AAABACADEAdeB R 2: ABACb R 3: AdEcb R 4: BCDDE R 5: AbeB R 6: Ced Relator 1 is a defining relator which was used to eliminate generator C. Swapped Relator 1 with Relator 6 and reduced the number of relators. Rewrote the relators by replacing E with C. Free reductions reduced the length of the current presentation from 66 to 60. The reduced presentation is: R 1: abaaabCDacdacd R 2: aabCDacdab R 3: AdCADCAdcBAAABAb R 4: BabaaabCDacdaDDC R 5: AbcB Do Aut 1 time(s): B->Ba C->Ca Do Aut 1 time(s): B->Ba D->AD 2 automorphism(s) reduced the length to 50. The presentation is currently: R 1: babCDcdacd R 2: bCDcdb R 3: daCADCdcBABAbA R 4: BababCDcdaDADCa R 5: AbAcB Rewrote the presentation using the substitution: abCD Saved the current presentation as: Presentation 5 Started with presentation 5, Length 50: R 1: ABABCDcdADaDCAb R 2: ABABCDcdAcaDAb R 3: ABCDcdAcdB R 4: BBCDcd R 5: ABCAb Relator 5 is a defining relator which was used to eliminate generator C. Rewrote the relators by replacing D with C. Free reductions reduced the length of the current presentation from 78 to 56. The reduced presentation is: R 1: BaCAbABcACaC R 2: ABBaCAbABcAAbABaCAb R 3: BaCAbABcAAbABcB R 4: BaBaCAbABc Do Aut 1 time(s): A->AB Do Aut 1 time(s): C->AC 2 automorphism(s) reduced the length to 48. The presentation is currently: R 1: CAABBcBACbC R 2: BBCAABBcBAABCAA R 3: CAABBcBAABBcaB R 4: aCAABBca Rewrote the presentation using the substitution: CAb Saved the current presentation as: Presentation 6 Started with presentation 6, Length 48: R 1: AABACCAbCCAAbCC R 2: AABACCAABcAbCC R 3: AABACbabbCC R 4: AABccbCC _______________________________________________________________________________________________ Finally, the program has obtained a presentation in which none of the relators is a primitive, or a proper power of a free generator. Since this is not a balanced presentation of a closed manifold, the program can't dualize the presentation, and so it starts looking for bandsums of pairs of relators that will reduce the length of the presentation. _______________________________________________________________________________________________ Checked whether any Relators are primitives or proper powers. . . Replaced Relator 1 with the following bandsum of Relator 1 and Relator 2: R 1: CbabCCA The current presentation is: R 1: CbabCCA R 2: AABACCAABcAbCC R 3: AABACbabbCC R 4: AABccbCC Do Aut 1 time(s): A->Ac B->CB Checked whether Relator 1 is a primitive or a proper power. . . Relator 1 is a primitive. Do Aut 1 time(s): A->Ac B->CB _______________________________________________________________________________________________ After performing the above automorphism on Relator 1, the total number of appearances of C and c in Relator 1 has been reduced to 1, and so Relator 1 has become a defining relator. _______________________________________________________________________________________________ Relator 1 is a defining relator which was used to eliminate generator C. Swapped Relator 1 with Relator 4 and reduced the number of relators. Free reductions reduced the length of the current presentation from 73 to 59. The reduced presentation is: R 1: AbabABAbabAbabAB R 2: AbabABABABAbabABAbabAAbabAB R 3: AbabABAbabAbabAB _______________________________________________________________________________________________ The program now discovers that Relator 3 is a cyclic conjugate of Relator 1 and so it deletes Relator 3 from the presentation. _______________________________________________________________________________________________ Deleted 1 duplicated relator(s) from the presentation to get: R 1: AbabABAbabAbabAB R 2: AbabABABABAbabABAbabAAbabAB Do Aut 1 time(s): A->Ab Do Aut 1 time(s): B->AB Rewrote the presentation using the substitution: ba Saved the current presentation as: Presentation 7 Started with presentation 7, Length 36: R 1: AAABBAbABBAbbbABBAbABB R 2: AABBAbABBAbABB Checked whether any Relators are primitives or proper powers. . . _______________________________________________________________________________________________ This presentation is now a balanced presentation of a closed orientable 3-manifold, and so the program can dualize the current presentation in order to look for simpler presentations. _______________________________________________________________________________________________ Dualized the current relators to get the following dual relators: R 1: ABABAABAABBAABA R 2: abaabaBAABAABaBAABAAB Checked whether any Relators are primitives or proper powers. . . Do Aut 1 time(s): B->Ba Do Aut 1 time(s): A->Ab 2 automorphism(s) reduced the length to 22. The presentation is currently: R 1: BAABaBAA R 2: aaBAABaBaBAABa Rewrote the presentation using the substitution: aB Saved the current presentation as: Presentation 8 Started with presentation 8, Length 22: R 1: AAABaaBABABaaB R 2: AAbAAbab Checked whether any Relators are primitives or proper powers. . . _______________________________________________________________________________________________ After verifying that neither of the relators in Presentation 8 are primitives, or proper powers of free generators, the program discovers that the Whitehead graph of Presentation 8 has a pair of separating vertices; so the program calls Level_Transformations() to see if it can transform Presentation 8 into a minimal length presentation without any pairs of separating vertices in its Whitehead graph. Level_Transformations() manages to find two such presentations, which the program saves as Presentation 9 and Presentation 10. _______________________________________________________________________________________________ Calling Level_Transformations() for the following presentation: R 1: AAABaaBABABaaB R 2: AAbAAbab Performed a level-transformation by sliding vertice(s): {B} along a path represented by: AA to obtain the presentation: R 1: AAABBAAABAAABBAA R 2: bbaaab Rewrote the presentation using the substitution: AB Saved the current presentation as: Presentation 9 Performed a level-transformation by sliding vertice(s): {B} along a path represented by: a to obtain the presentation: R 1: AABaaaBBBaaaB R 2: AAbAAAbbA Rewrote the presentation using the substitution: aB Saved the current presentation as: Presentation 10 Performed a level-transformation by sliding vertice(s): {b} along a path represented by: A to obtain the presentation: R 1: AABaaaBBBaaaB R 2: AAbAAAbbA Rewrote the presentation using the substitution: aB Performed a level-transformation by sliding vertice(s): {b} along a path represented by: aa to obtain the presentation: R 1: AAABBAAABAAABBAA R 2: bbaaab Rewrote the presentation using the substitution: AB Started with presentation 9, Length 22: R 1: AAAAABBAAABAAABB R 2: AAABBB Checked whether any Relators are primitives or proper powers. . . Dualized the current relators to get the following dual relators: R 1: AABAAAABAAABAA R 2: AABABAAB Do Aut 3 time(s): B->Ba Do Aut 1 time(s): A->Ab _______________________________________________________________________________________________ The program is checking whether Relator 1 is a primitive by attempting to reduce Relator 1 to a defining relator for one of the generators. Note: Here the program recognizes that the automorphism B->Baaa can be performed, and it performs this automorphism instead of performing B->Ba 3 times. (Doing this can make performing automorphisms significantly more efficient when generators start appearing with large exponents.) _______________________________________________________________________________________________ Checked whether any Relators are primitives or proper powers. . . Relator 1 is a primitive. _______________________________________________________________________________________________ The program has just recognized that this is a 2-generator, 2-relator presentation in which Relator 1 is a primitive, so the program calls the routine Lens_Space_D(), to check whether the manifold in question might be a lens space. Lens_Space_D() verifies that the manifold is indeed a lens space, and then determines that the lens space is homeomorphic to L(18,5). _______________________________________________________________________________________________ The current presentation is a two generator, two relator presentation of a closed manifold M, for which one of the relators is a primitive. Thus M is a lens space. In particular, M is the lens space L(18,5). _______________________________________________________________________________________________ The program next realizes that the current relators were obtained by dualizing and reverts to the original set of relators, which it rewrites and then discovers is already on file as Presentation 8. So, the program appends a message to Presentation 8 indicating that Presentation 8 represents the lens space L(18,5), and then the program stops running this example because it has recognized all of the components of the original manifold. _______________________________________________________________________________________________ Rewrote the presentation using the substitution: AB _______________________________________________________________________________________________ Finally, the program prints the following report. _______________________________________________________________________________________________ <------------------------------------ REPORT ----------------------------------------> Presentation 1 of Component 1: 12 Generator(s) Length 59 From Presentation 1 IP vertex 'a' and vertex 'B' separate the diagram. R 1) AAAAABCDEAFGHIJ R 2) AgeFE R 3) BHchg R 4) AjdJ R 5) BHbK R 6) CIch R 7) CIdi R 8) DJdi R 9) EfLF R 10) Alg R 11) Bge R 12) AK R 13) Fj Presentation 2 of Component 1: 12 Generator(s) Length 59 From Presentation 1 Lt via level transformations of sep_pairs. R 1) AAAAABCDEAFGHIJ R 2) AFEfKg R 3) ABhbL R 4) AgeFE R 5) BHchg R 6) AjdJ R 7) CIch R 8) CIdi R 9) DJdi R 10) Bge R 11) Fj R 12) K R 13) L Presentation 3 of Component 1: 8 Generator(s) Length 46 From Presentation 2 FP R 1) AAAABCDEAFefGH R 2) AGChcb R 3) AFFEb R 4) AdHDE R 5) CgefG R 6) CHdh R 7) BfE R 8) Ad R 9) Bg Presentation 4 of Component 1: 5 Generator(s) Length 35 From Presentation 3 FP R 1) AAABACADEAdeB R 2) ABACb R 3) AdEcb R 4) BCDDE R 5) AbeB R 6) Ced Presentation 5 of Component 1: 4 Generator(s) Length 50 From Presentation 4 FP R 1) ABABCDcdADaDCAb R 2) ABABCDcdAcaDAb R 3) ABCDcdAcdB R 4) BBCDcd R 5) ABCAb Presentation 6 of Component 1: 3 Generator(s) Length 48 From Presentation 5 FP R 1) AABACCAbCCAAbCC R 2) AABACCAABcAbCC R 3) AABACbabbCC R 4) AABccbCC Presentation 7 of Component 1: 2 Generator(s) Length 36 From Presentation 6 DD R 1) AAABBAbABBAbbbABBAbABB R 2) AABBAbABBAbABB Presentation 8 of Component 1: 2 Generator(s) Length 22 From Presentation 7 BC vertex 'A' and vertex 'a' separate the diagram. R 1) AAABaaBABABaaB R 2) AAbAAbab Presentation 9 of Component 1: 2 Generator(s) Length 22 From Presentation 8 Lt presents the Lens space L(18,5). R 1) AAAAABBAAABAAABB R 2) AAABBB Presentation 10 of Component 1: 2 Generator(s) Length 22 From Presentation 8 Lt via level transformations of sep_pairs. R 1) AAABBBAAABaaB R 2) AAABAAABB The program performed 32 automorphism(s), examined 12 bandsum(s) and examined 10 diagram(s). Left to do 69. _______________________________________________________________________________________________ Note that in the line above, the program reports that it examined 12 bandsums while in the MicroPrint log, it only reported forming 1 bandsum. The discrepency arises because the program may examine several possible bandsums before it decides which bandsum it will actually perform. So, in this case, the program considered 12 possibilities before it chose the bandsum which it reported performing. _______________________________________________________________________________________________