J b $.Heegaard_Program_7/20/98APPLJOB?N@EJEsXKm 0! 6+/p/NhO ` A"L AHeegaard_Program_7/20/98/APAPPLJOB?!>$.<(N /"< o H>0C&x*<J8/fH" N*L`LCPEG8,HF1`H؅bx*<,<J8/fof a closed manifold M, in which one of the relators is a primitive. Thus M is a lens space. In particular, M is the lens space L(%lu,%lu). The current presentation is a two generator, two relator presentation of a closed manifold M, in which one of the relators is a primitive. Thus M is a lens space. In particular, M is the lens space L(%lu,%lu). Swapped the presentation with a copy of the previous presentation to get: Swapped the presentation with a copy of the previous presentation to get: %lu automorphism(s) reduced the length to %lu. The presentation is currently: %lu automorphism(s) reduced the length to %lu. The presentation is currently: The current set of relators has minimal length of %lu. The current set of relators has minimal length of %lu. The current Presentation is: The current Presentation is: The presentation of summand %d is: Saved this presentation as: Presentation %u The presentation of summand %d is: Saved this presentation as: Presentation %u Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6d ER The presentation of summand %d is currently: The presentation of summand %d is currently: The presentation of summand %d is: Saved this presentation as: Presentation %u The presentation of summand %d is: Saved this presentation as: Presentation %u Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6u ER Relator %d of the preceding presentation is a primitive or proper power whose deletion creates empty relators in the presentation. This means the manifold 'splits' as a connected sum. The program will look for presentations corresponding to each summand. Relator %d of the preceding presentation is a primitive or proper power whose deletion creates empty relators in the presentation. This means the manifold 'splits' as a connected sum. The program will look for presentations corresponding to each summand. A relator in the preceding dual presentation freely reduces to an empty relator. This means the manifold 'splits' as a connected sum. The program will look for presentations corresponding to each summand. A relator in the preceding dual presentation freely reduces to an empty relator. This means the manifold 'splits' as a connected sum. The program will look for presentations corresponding to each summand. Stopping because the program cannot deal with any more summands. Sorry! There is an annulus in diagram %u. There is an annulus in diagram %u. Relator %d is trivial. Deleted generator %c from the presentation. Swapped Relator %d with Relator %d and reduced the number of relators. Relator %d is trivial. Deleted generator %c from the presentation. Swapped Relator %d with Relator %d and reduced the number of relators. Rewrote the relators by replacing %c with %c. Rewrote the relators by replacing %c with %c. There is a 'valence-two' annulus in diagram %u. There is a 'valence-two' annulus in diagram %u. Stopping because the program cannot deal with any more summands. Sorry! The presentation has reduced to an empty presentation. The presentation has reduced to an empty presentation. A generator no longer appears in the relators. The manifold therefore has a 'handle' of some kind. A generator no longer appears in the relators. The manifold therefore has a 'handle' of some kind. %d generators no longer appear in the relators. The manifold therefore has %d 'handles' of some kind. %d generators no longer appear in the relators. The manifold therefore has a 'handle' of some kind. The presentation is currently: Found Presentation %u and Presentation %u. The presentation is currently: Found Presentation %u and Presentation %u. Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6d MG Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6d MG The current dual presentation contains %d empty relator(s): The current dual presentation contains %d empty relator(s): Deleted an empty relator created by a bandsum. The presentation is currently: Deleted an empty relator created by a bandsum. The presentation is currently: Presentation %d is too long! Presentation %d does not have minimal length. The program will only find level transformations for presentations that have minimal length. Presentation %d does not have a connected Whitehead graph. The program will only find level transformations for presentations with connected graphs! Out of memory. Sorry! Presentation %d does not have minimal length. Presentation %d does not have minimal length. There is an automorphism which reduces the number of appearances of generator %c. There is an automorphism which reduces the number of appearances of generator %c. Performed the following level-transformation: Performed the following level-transformation: The presentation is currently: The presentation is currently: The program found %lu level transformations(s). The program found %lu level transformations(s). Total presentations with no pairs of separating vertices: %u. Total pseudo-minimal presentations: %u. (Ignore this number if the manifold is not closed.) Total reducible presentations: %u. (Ignore this number if the manifold is not closed.) Unable to determine if any presentations in the orbit are realizable. None of the presentations in the orbit are realizable. All presentations in the orbit are realizable. Unable to determine if any presentations in the orbit are realizable. Some kind of error occured. The program found both realizable and unrealizable presentations in the orbit!!!?? Presentations with no pairs of separating vertices: {%3d,{%3d}.%3d,%3d}. SAVE THIS LIST IN THE FILE 'Heegaard_Results' ? HIT 'y' OR 'n'. Presentations which are pseudo-minimal: {%3d,{%3d}.%3d,%3d}. SAVE THIS LIST IN THE FILE 'Heegaard_Results' ? HIT 'y' OR 'n'. Presentations which are reducible: {%3d,{%3d}.%3d,%3d}. Presentations whose duals are Standard: {%3d,{%3d}.%3d,%3d}. SAVE THIS LIST IN THE FILE 'Heegaard_Results' ? HIT 'y' OR 'n'. PLEASE SELECT ONE OF THE FOLLOWING ALTERNATIVES. 0) Save no 'level' presentations. 1) Save only 'level' presentations which have no pairs of separating vertices. 2) Save only 'level' presentations which are pseudo-minimal. 3) Save only 'level' presentations which are reducible. 4) Save all 'level' presentations. Any presentations saved will appear in 'Heegaard_Results'. HIT 0,1,2,3,OR 4. %d -> %d. %d -> %d. %d -> %d. %d -> %d. %d -> %d. %d -> %d. A Torsion value is greater than 4,294,967,290. This is an overflow. Sorry! A Torsion value is greater than 4,294,967,290. This is an overflow. Sorry! The homology is trivial. The homology is: Z%lu x The homology is: Z x Z x Z%lu This may be a Wirtinger presentation of a knot. TRY DEHN-FILLING ON THIS HYPOTHETICAL KNOT ? HIT 'y' OR 'n'. This may be a Wirtinger presentation of a %d-component link. TRY DEHN-FILLING(S) ON THIS HYPOTHETICAL LINK ? HIT 'y' OR 'n'. NOTE: TO SKIP FILLING A PARTICULAR BOUNDARY COMPONENT HIT 'return'. NOTE: TO SKIP DOING THIS FILLING HIT 'return'. This presentation now has %d relators. The program cannot accept more than %d relators. So no fillings are possible. Sorry! So no more fillings are possible. Sorry! Component %d: Meridian: %s Longitude: %s Please specify the slope M/L of the surgery on this component or hit 'return'. 1) ENTER THE VALUE OF M AND HIT 'return'. The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign!%d2) ENTER THE VALUE OF L AND HIT 'return'. The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign! %d Both M and L are zero! Please specify different surgery coefficients for this component! These coefficients give a new relator which will probably be too long! Please specify new surgery coefficients for this component. Tried Dehn-filling on: %s Tried %d/%d surgery with meridian: %s and longitude: %s. These coefficients give a new relator which is too long! Please specify new surgery coefficients for this component. Exponent surgery is not possible for this presentation. Sorry! Exponent surgery is not possible for this presentation. Sorry! Exponent surgery is not possible for this presentation. Sorry! Exponent surgery is not possible for this presentation. Sorry! The initial presentation did not have minimal length; one automorphism reduced its length to %lu. The initial presentation did not have minimal length; %lu automorphisms reduced its length to %lu. The presentation is currently: GENERATOR '%c' APPEARS WITH EXPONENTS P = %u, Q = %u, AND P+Q = %u. REPLACE P WITH: The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign!%dREPLACE Q WITH: The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign! %d Both P and Q are zero! P and Q are not relatively prime! Dividing P and Q by their GCD %u. Setting P = %d, Q = %d. GENERATOR '%c' APPEARS WITH EXPONENTS P = %u, AND Q = %u. REPLACE P WITH: The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign!%dREPLACE Q WITH: The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign! %d Both P and Q are zero! P and Q are not relatively prime! Dividing P and Q by their GCD. Setting P = %d, Q = %d. GENERATOR '%c' APPEARS ONLY WITH EXPONENT P = %u. REPLACE P WITH: The first character must be a digit, '+' or '-' sign! Each character must be a digit or a leading '+' or '-' sign!%d Out of memory! Can't do exponent surgery. Relator %d will be too long! TRY DIFFERENT EXPONENTS ? HIT 'y' OR 'n'. Relator %d will be too long! TRY DIFFERENT EXPONENTS ? HIT 'y' OR 'n'. The surgered presentation is: The surgered presentation is: This presentation has length %ld This presentation has length %ld This presentation is too long!!and freely reduces to length %lu.and freely reduces to length %lu.and is freely reduced.and is freely reduced. There must be at least one non-empty relator!! NumRelators = %d, NumGenerators = %d The rewritten initial presentation is: The rewritten initial presentation is: Memory Error. Sorry! Surgery is not possible for this presentation. Sorry! Surgery is not possible for this presentation. Sorry! Surgery is not possible for this presentation. Sorry! The initial presentation did not have minimal length; one automorphism reduced its length to %lu. The initial presentation did not have minimal length; %lu automorphisms reduced its length to %lu. Memory error. Sorry! Cutting disk surgery gives a way to obtain different Heegaard diagrams by changing the way in which the ends of edges meeting pairs of inverse vertices are identified. By choosing values for the offsets, you can obtain the diagrams of all 3-manifolds which have Heegaard diagrams with Whitehead graphs isomorphic to the Whitehead graph of the initial diagram. The presentation is currently: The following arrays give the 'valence' of each generator and the current values of the offsets. Gens:%5d Vals:%5u OSet:%5u Please enter new non-negative values for the offsets of each generator. Generator %2d, Valence %5u, Current offset %5u. Change offset to ? %u TRY ANOTHER SURGERY ON THIS PRESENTATION ? HIT 'y' OR 'n'. The surgered presentation is: The surgered presentation is: This presentation has length %ld This presentation has length %ld This presentation is too long!! TRY ANOTHER SURGERY ON THIS PRESENTATION ? HIT 'y' OR 'n'. and freely reduces to length %lu.and freely reduces to length %lu.and is freely reduced.and is freely reduced. There must be at least one non-empty relator!! TRY ANOTHER SURGERY ON THIS PRESENTATION ? HIT 'y' OR 'n'. NumRelators = %d, NumGenerators = %d The rewritten initial presentation is: USE THIS PRESENTATION ? HIT 'y' OR 'n'. The rewritten initial presentation is: Memory Error. Sorry! Memory error. Sorry! This presentation now has %d relators. The program cannot accept more than %d relators. Sorry! Memory error. Sorry! Relator %d of this presentation is longer than %d. This is too long for the program to handle. Sorry! LIST THE NUMBER OF ACTIVE DESCENDANTS OF EACH PRESENTATION ? HIT 'y' OR 'n'. Pres : NumDes:%4d Pres :%4d NumDes:%4d CURRENT PRESENTATIONS RANGE FROM NUMBER 1 TO NUMBER %u. ENTER THE NUMBER OF A PRESENTATION, WHOSE PROPER DESCENDANTS YOU WANT THE PROGRAM TO STOP SEARCHING, AND HIT 'return'. %d Made %d currently active proper descendant(s) of Presentation %d inactive. No proper descendants of Presentation %d are currently active. DO MORE PRUNNING ? HIT 'y' OR 'n'. Memory is getting rather low! It may be wise to save and quit! The program has used %lu K bytes of memory for storing presentations. This exceeds the current limit of %lu K bytes reserved for this purpose. HIT 'c' TO CONTINUE RUNNING THIS EXAMPLE. HIT 't' TO TERMINATE THIS RUN. HIT 'v' TO REVIEW THE PRESENTATIONS NOW IN MEMORY. ENTER A NEW UPPER LIMIT ON THE AMOUNT OF MEMORY TO BE USED (in K) AND HIT 'return'. %lu The number of bytes available must exceed the number already used!! ENTER THE MaxSavedLength OF A PRESENTATION. %ld HIT 'v' TO REVIEW THESE SORTED PRESENTATIONS. HIT 's' TO SAVE THESE SORTED PRESENTATIONS TO THE FILE 'Heegaard_Results'. OR HIT ANY OTHER KEY TO CONTINUE. <------------------------------------ REPORT ----------------------------------------> Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Use the 'd','i','n','p' and 's' keys to: view diagrams, save a presentation in 'Input_Presentations', see the next presentation, see the previous presentation, or save a presentation in 'Heegaard_Results'. The initial presentation was: %s Presentation %d of Summand %u: %d Generator(s) Length %lu From Presentation %u DRIPLS1GS3FPBCPMPMBCCFERErNCMGPPLtLTA2 The presentation dual to presentation %d 'split' into presentations %u through %u of M which correspond to summands %u through %u of M. 'split' into presentations %u through %u corresponding to summands %u through %u of M. presents a Lens space of the form L(%lu,Q). presents the 3-Sphere. presents a Lens space of the form L(%lu,1). presents the 3-Sphere. the diagram is not connected. presents S1 X S2. presents %d copies of S1 X S2. presents S1 X D2. presents %d copies of S1 X D2. presents S1 X ?2. presents %d copies of S1 X ?2. presents: 1 copy of I X D2, %d copies of I X D2, 1 copy of S1 X S2, %d copies of S1 X S2, and 1 copy of S1 X D2.and %d copies of S1 X D2. The program was not able to unambiguously determine the boundary components. presents: 1 copy of I X D2, %d copies of I X D2, 1 copy of S1 X S2, %d copies of S1 X S2, and 1 copy of S1 X D2.and %d copies of S1 X D2. presents S1 X S2. presents the 3-Sphere. presents the Lens space L(%lu,%lu). dual of presentation %u is pseudo-minimal. Vertex '%c' and vertex '%c' separate. vertex '%c' and vertex '%c' separate the diagram. Vertices '%c' and '%c' separate the diagram. The component consisting of vertice(s) {%c,%c} lies in an annulus which swallows the component and otherwise follows the curve: There exists an annulus which swallows vertice(s) {%c,%c} and otherwise follows the curve: the diagram is not unique because there is a generator which appears with only one exponent and that exponent is greater than 6. the diagram is not unique because there is a generator which appears only with exponent 5. the diagram is not unique because there is a generator which appears only with exponent 3 or only with exponent 4 and this exponent occurs more than once. the diagram is not unique because there is a generator which appears with only one exponent, either 3,4 or 6, and a needed symmetry does not exist. is a duplicate of presentation %d of summand %u. dual of presentation %u is pseudo-minimal. via level transformations of sep_pairs, is quasi-pseudo-minimal. via level transformations of sep_pairs. via general level transformations, is quasi-pseudo-minimal. via general level transformations. via level transformations of sep_pairs and dual of presentation %u is pseudo-minimal. via general level transformations and dual of presentation %u is pseudo-minimal. dual of presentation %u is pseudo-minimal. is quasi-pseudo-minimal. R%3u) Saved Presentation %d in 'Heegaard_Results'. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to start scrolling. Saved Presentation %d in 'Heegaard_Results'. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. The program performed %lu automorphism(s), examined %ld bandsum(s), examined %ld diagram(s), and dualized %lu diagram(s). Left to do %u. These results are printed in the file 'Heegaard_Results'. HIT 'v' TO REVIEW THESE PRESENTATIONS. HIT 's' TO SAVE THESE PRESENTATIONS TO THE FILE 'Heegaard_Results'. OR HIT ANY OTHER KEY TO CONTINUE. <------------------------------------ REPORT ----------------------------------------> Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Use the 'd','i','n','p' and 's' keys to: view diagrams, save a presentation in 'Input_Presentations', see the next presentation, see the previous presentation, or save a presentation in 'Heegaard_Results'. The initial presentation was: %s Presentation %d of Summand %u: %d Generator(s) Length %lu From Presentation %u DRIPLS1GS3FPBCPMPMBCCFERErNCMGPPLtLTA2 The presentation dual to presentation %d 'split' into presentations %u through %u of M which correspond to summands %u through %u of M. 'split' into presentations %u through %u corresponding to summands %u through %u of M. presents a Lens space of the form L(%lu,Q). presents the 3-Sphere. presents a Lens space of the form L(%lu,1). presents the 3-Sphere. the diagram is not connected. presents S1 X S2. presents %d copies of S1 X S2. presents S1 X D2. presents %d copies of S1 X D2. presents S1 X ?2. presents %d copies of S1 X ?2. presents: 1 copy of I X D2, %d copies of I X D2, 1 copy of S1 X S2, %d copies of S1 X S2, and 1 copy of S1 X D2.and %d copies of S1 X D2. The program was not able to unambiguously determine the boundary components. presents: 1 copy of I X D2, %d copies of I X D2, 1 copy of S1 X S2, %d copies of S1 X S2, and 1 copy of S1 X D2.and %d copies of S1 X D2. presents S1 X S2. presents the 3-Sphere. presents the Lens space L(%lu,%lu). dual of presentation %u is pseudo-minimal. Vertex '%c' and vertex '%c' separate. vertex '%c' and vertex '%c' separate the diagram. Vertices '%c' and '%c' separate the diagram. The component consisting of vertice(s) {%c,%c} lies in an annulus which swallows the component and otherwise follows the curve: There exists an annulus which swallows vertice(s) {%c,%c} and otherwise follows the curve: the diagram is not unique because there is a generator which appears with only one exponent and that exponent is greater than 6. the diagram is not unique because there is a generator which appears only with exponent 5. the diagram is not unique because there is a generator which appears only with exponent 3 or only with exponent 4 and this exponent occurs more than once. the diagram is not unique because there is a generator which appears with only one exponent, either 3,4 or 6, and a needed symmetry does not exist. is a duplicate of presentation %d of summand %u. dual of presentation %u is pseudo-minimal. via level transformations of sep_pairs, is quasi-pseudo-minimal. via level transformations of sep_pairs. via general level transformations, is quasi-pseudo-minimal. via general level transformations. via level transformations of sep_pairs and dual of presentation %u is pseudo-minimal. via general level transformations and dual of presentation %u is pseudo-minimal. dual of presentation %u is pseudo-minimal. is quasi-pseudo-minimal. R%3u) Saved Presentation %d in 'Heegaard_Results'. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to start scrolling. Saved Presentation %d in 'Heegaard_Results'. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Hit 'd' to see the diagram of presentation %d. Hit 'i' to save presentation %d to 'Input_Presentations'. Hit 'n' to see the next presentation. Hit 'p' to see the previous presentation. Hit 's' to save presentation %d to 'Heegaard_Results'. Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. The program performed %lu automorphism(s), examined %ld bandsum(s), examined %ld diagram(s), and dualized %lu diagram(s). Left to do %u. These results are printed in the file 'Heegaard_Results'. HIT 'v' TO REVIEW THESE PRESENTATIONS. HIT 's' TO SAVE THESE PRESENTATIONS TO THE FILE 'Heegaard_Results'. OR HIT ANY OTHER KEY TO CONTINUE. The program was unable to determine the boundary components of this manifold. The program was unable to determine the boundary components of 'summand' %u of M. This manifold is closed. 'Summand' %u of M is closed. This manifold has one boundary component of genus %u. 'Summand' %u of M has one boundary component of genus %u. This manifold has %u boundary components of genus %u. 'Summand' %u of M has %u boundary components of genus %u. This manifold has the following boundary components: 'Summand' %u of M has the following boundary components: %2u component of genus %2u. %2u components of genus %2u. Presentation %d is not realizable. Presentation %d is not realizable. This presentation, obtained from presentation %d, is not realizable. This presentation obtained, from presentation %d, is not realizable. R%3u: Started with Presentation %d of Summand %d, Length %lu: Started with Presentation %d of Summand %d, Length %lu: R%3u: R%3u: Free reductions reduced the length of the current presentation from %lu to %lu. The reduced presentation is: Free reductions reduced the length of the current presentation from %lu to %lu. The reduced presentation is: Dualized the current relators to get the following dual relators: Dualized the current relators to get the following dual relators: Replaced Relator %u with the following bandsum of Relator %u and Relator %u. Delta Length = %ld. Replaced Relator %u with the following bandsum of Relator %u and Relator %u. Delta Length = %ld. And then swapped Relator %u and Relator 1. And then swapped Relator %u and Relator 1. The current presentation is: The current presentation is: Looking for level-transformations of: Looking for level-transformations of: Vertices %c and %c form a Type %u separating pair. Vertices %c and %c form a Type %u separating pair. Performed a level-transformation by sliding vertice(s): {%s} along a path represented by: %s to obtain the presentation: Performed a level-transformation by sliding vertice(s): {%s} along a path represented by: %s to obtain the presentation: Do Aut %u time(s): Do Aut %u time(s): %6lu) %6lu) %c->%c%c %c->%c%c %c->%c%c%c %c->%c%c%c %c->%c%c %c->%c%c %c->%c%c %c->%c%c %c->%c%c %c->%c%c %c->%c%c%c %c->%c%c%c R %2u: %s The 'Dual' Relators of Diagram %d are: R%3u: SAVE A COPY OF THIS DATA IN 'Heegaard_Results' ? HIT 'y' OR 'n'. HIT 'P' TO PRINT THE DIAGRAM. HIT 'm' TO RETURN TO DIAGRAM %d. HIT 'v' TO REVIEW PRESENTATION %d. HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. HIT 'n' TO SEE THE NEXT DIAGRAM. HIT 'p' TO SEE THE PREVIOUS DIAGRAM. HIT 'q' TO QUIT VIEWING DIAGRAMS. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. The 'Out' Relators of Diagram %d are: R%3u: SAVE A COPY OF THIS DATA IN 'Heegaard_Results' ? HIT 'y' OR 'n'. HIT 'P' TO PRINT THE DIAGRAM. HIT 'm' TO RETURN TO DIAGRAM %d. HIT 'v' TO REVIEW PRESENTATION %d. HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. HIT 'n' TO SEE THE NEXT DIAGRAM. HIT 'p' TO SEE THE PREVIOUS DIAGRAM. HIT 'q' TO QUIT VIEWING DIAGRAMS. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. SLR [%2u][%2u]: %s Memory Error. Sorry! This is an empty presentation. There is nothing to display. Unable to display diagram %d. Sorry! NOTE! Presentation 1 does not have minimal length. The program will only display diagrams of minimal length presentations. Presentation 2 should give a diagram of a minimal length version of presentation 1. Unable to display diagram %d. Sorry! Unable to display diagram %d. Sorry! HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. Displaying Diagrams. . . REVIEW ALL PRESENTATIONS AVAILABLE ? HIT 'y' OR 'n'. CONTINUE TO REVIEW PRESENTATIONS ? HIT 'y' OR 'n'. a) SHOW ALL DIAGRAMS, b) SHOW ALL DIAGRAMS THAT ARE CONNECTED AND HAVE NO PAIRS OF SEPARATING VERTICES, c) OR SHOW YOUR CHOICE OF A PARTICULAR DIAGRAM ? HIT 'a','b', OR 'c' ENTER A DIAGRAM FROM 1 TO %u THAT YOU WANT TO SEE AND HIT 'return'. Except for diagram %d which is a diagram of S1 X S2 or S1 X D2 and is empty. Except for diagrams: {%d,%d,%d}. These are diagrams of S1 X S2 (s) or S1 X D2 (s) and are empty. %d Memory Error. Sorry! This is an empty presentation. There is nothing to display. Unable to display diagram %d. Sorry! NOTE! Presentation 1 does not have minimal length. The program will only display diagrams of minimal length presentations. Presentation 2 should give a diagram of a minimal length version of presentation 1. HIT ANY KEY TO CONTINUE. Unable to display diagram %d. Sorry! Unable to display diagram %d. Sorry! Memory Error. Sorry! Unable to display diagram %d. Sorry! NOTE! Presentation 1 does not have minimal length. The program will only display diagrams of minimal length presentations. Presentation 2 should give a diagram of a minimal length version of presentation 1. HIT ANY KEY TO SEE THE NEXT DIAGRAM. Unable to display diagram %d. Sorry! Unable to display diagram %d. Sorry! There are no diagrams which are connected and without pairs of separating vertices! CONTINUE DISPLAYING DIAGRAMS ? HIT 'y' OR 'n'. Input_Presentationsr+ Unable to open the file 'Input_Presentations'. Memory error. Sorry! Memory error. Sorry! Please enter a name by which the program can refer to this presentation, and then hit 'return'. Or just hit 'return' to skip saving this presentation. SAVE THIS PRESENTATION AS: Hit 'return' to stop reviewing this report. Hit the 'space-bar' to alternately stop and start scrolling. Use the 'd','i','n','p' and 's' keys to: view diagrams, save a presentation in Input_Presentations, see the next presentation, see the previous presentation, or save a presentation in Heegaard_Results. Input_Presentations already contains a presentation with this identifier! %s %s Saved Presentation %d in 'Input_Presentations' as: %s HEEGAARDHeegaard_Resultsa+ Unable to open the file 'Heegaard_Results'. ENTER THE MaxSavedLength OF A PRESENTATION. %ld Pres %d does not have minimal length. Whitehead_Graph returns an error on pres %d. Looking for new presentations obtained by reducing the above presentations to minimal length. . . From %4d, Delta Length = %5ld, Auts = %5ld length > max in a bandsum! Error in forming a bandsum. Relators are probably too long. Error in forming a bandsum. Relators are probably too long. %d duplicated or empty relators. NOTE: This modification may change the homeomorphism type of M. NOTE: This modification may change the homeomorphism type of M. The modified presentation is: The modified presentation is: Unable to determine whether the presentation is realizable. Unable to determine whether the presentation is realizable. Suggest rerunning the example and allowing the program to delete all primitives. Suggest rerunning the example and allowing the program to delete all primitives. However, the realization is not unique because an annulus exists. However, the realization is not unique because an annulus exists. Looking for other diagrams. . . . Saved the current presentation as: Presentation %u. Saved the current presentation as: Presentation %u. Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6u DRIPLS1GS3FPBCPMPMBCCFERErNCMGPPLtLTA2 Memory Error. Sorry! Presentation %d is realizable. Presentation %d is realizable. But the realization is not unique. But the realization is not unique. NOTE: The program is investigating one diagram that realizes the initial presentation. Subsequent results may not apply to all realizations. NOTE: The program is investigating one diagram that realizes the initial presentation. Subsequent results may not apply to all realizations. Unable to delete a primitive. Unable to delete a primitive. NOTE: The program is investigating one diagram that realizes the initial presentation. Subsequent results may not apply to all realizations. NOTE: The program is investigating one diagram that realizes the initial presentation. Subsequent results may not apply to all realizations. The program found a relator which is a primitive and deleted its consequences from the presentation. The program found a relator which is a primitive and deleted its consequences from the presentation. The program found a relator which is a proper power and deleted its consequences from the presentation. The program found a relator which is a proper power and deleted its consequences from the presentation. This manifold is a lens space. This manifold is a lens space. This manifold is a connected sum. This manifold is a connected sum. The initial presentation is not realizable. The initial presentation is not realizable. A copy of the current presentation is already on file as presentation %d. A copy of the current presentation is already on file as presentation %d. The current presentation is a duplicate of presentation %d of summand %d. The current presentation is a duplicate of presentation %d of summand %d. Caution!!! All the presentations in memory, except the one you rerun, will be vaporized unless you save them now! HIT 'v' TO REVIEW THESE PRESENTATIONS. HIT 's' TO SAVE THESE PRESENTATIONS TO THE FILE 'Heegaard_Results'. OR HIT ANY OTHER KEY TO CONTINUE. CONTINUE TO REVIEW PRESENTATIONS ? HIT 'y' OR 'n'. RETRIEVE THE LAST PRESENTATION USED FOR CUTTING DISK SURGERY ? HIT 'y' OR 'n'. Memory Error. Sorry! The initial presentation is: %s RETRIEVE THE LAST PRESENTATION USED FOR EXPONENT SURGERY ? HIT 'y' OR 'n'. Memory Error. Sorry! The initial presentation is: %s RETRIEVE THE LAST WIRTINGER PRESENTATION OF A KNOT OR LINK ? HIT 'y' OR 'n'. Memory Error. Sorry! The initial presentation is: %s Surgery is not possible. Sorry! The surgered presentation is: ENTER A PRESENTATION FROM 0 TO %u THAT YOU WANT RERUN AND HIT 'return'. Except for presentation %d which is a presentation of S1 X S2 or S1 X D2 and is empty. Except for presentations: {%d,%d,%d}. These are presentations of S1 X S2 (s) or S1 X D2 (s) and are empty. %d Rerunning the original presentation. Memory Error. Sorry! Rerunning presentation %d. A rerun of Presentation Memory Error. Sorry!Input_Presentationsr+ Unable to open the file 'Input_Presentations'. This is the file 'Input_Presentations'. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling. Unable to display the file 'Heegaard_Results'. This is the file 'Heegaard_Results'. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling.Heegaard_Helpr Unable to open the Help file 'Heegaard_Help'. Please locate the file 'Heegaard_Help' and put it in the same folder with the program. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling. Note: Hit 'return' to stop reviewing this file. Hit the 'space-bar' to alternately stop and start scrolling.Input_Presentationsr+ Unable to open the file 'Input_Presentations'. Please locate the file 'Input_Presentations', make sure it is closed, and place it in the same folder that the program is in. PLEASE INDICATE WHICH PRESENTATION YOU WANT USED BY ENTERING ITS IDENTIFIER: Unable to find this presentation in the file 'Input_Presentations'. REVIEW THE FILE 'Input_Presentations' ? HIT 'y' OR 'n'. Unable to find any non-trivial relators in this presentation! R%3u) Please check relator %2u. A relator can only contain upper and lower case letters! Please make sure each relator is on its own line! And the presentation is terminated by an empty line! Relator %d is too long!! Memory Error. Sorry! The program will not accept any more than %d relators. Memory Error. Sorry! PLEASE ENTER THE RELATORS. (Note: Hit 'return' twice to terminate entry of the presentation.) R%3u) A relator can contain only upper and lower case letters! No blanks or trailing spaces! PLEASE ENTER RELATOR %2u. Memory Error. Sorry! The program will not accept more than %d relators. Please enter a name by which the program can refer to this presentation, and then hit 'return'. SAVE THIS PRESENTATION AS: DISCARD ALL BUT THE INITIAL PRESENTATION ? HIT 'y' OR 'n'. This presentation may be too long! Unable to display its diagram. Sorry! The presentation has reduced to the empty presentation. There is no diagram to display. NOTE! The initial presentation does not have minimal length. The diagram you see will correspond to the following rewritten minimal length version of the initial presentation. Note: a generator disappeared from the original presentation. Note: %d generators disappeared from the original presentation. The missing generator(s) will not appear in the Heegaard diagram. A copy of this presentation will be saved in the file 'Heegaard_Results'. The minimal length presentation is: The minimal length presentation is: This presentation has length %ld This presentation has length %ld HIT ANY KEY TO SEE THE HEEGAARD DIAGRAM. This presentation may be too long! Unable to display its diagram. Sorry! The program is currently forming only length reducing bandsums. The program is currently forming all possible bandsums. The program is not currently forming any bandsums. CREATE NEW DIAGRAMS BY FORMING BANDSUMS ? HIT 'y' OR 'n'. HIT 'a' TO FORM ALL POSSIBLE BANDSUMS. HIT 'r' TO FORM ONLY LENGTH REDUCING BANDSUMS. HIT 'n' TO FORM NO BANDSUMS. The program is currently trying to delete all primitive relators. CONTINUE TO DELETE ALL PRIMITIVE RELATORS ? HIT 'y' OR 'n'. The program is currently deleting only primitive relators of length 1 and 2. The program is not currently attempting to delete any primitive relators. CHANGE TO DELETE ALL PRIMITIVE RELATORS ? HIT 'y' OR 'n'. DELETE ALL PRIMITIVE RELATORS ? HIT 'y' OR 'n'. DELETE PRIMITIVE RELATORS OF LENGTH 1 AND LENGTH 2 ? HIT 'y' OR 'n'. The program is currently trying to make broad use of level-transformations. CONTINUE TO MAKE BROAD USE OF LEVEL-TRANSFORMATIONS ? HIT 'y' OR 'n'. The program is currently trying to make only limited use of level-transformations. CHANGE STRATEGY TO MAKE BROAD USE OF LEVEL-TRANSFORMATIONS ? HIT 'y' OR 'n'. TURN Micro_Printing ON ? HIT 'y' OR 'n'. HIT 'b' TO MICRO_PRINT TO BOTH THE SCREEN AND 'Heegaard_Results'. HIT 's' TO MICRO_PRINT ONLY TO THE SCREEN. HIT 'o' TO TURN MICRO_PRINTING COMPLETELY OFF. This is an empty presentation!! Presentation %d is too long! Presentation %d does not have minimal length. The program will only find level transformations for presentations that have minimal length. PRINT STABILIZER INFO INTO 'Heegaard_Results' ? HIT 'y' OR 'n'. Listing members of the orbit of presentation 1 under level transformations. . . We have run out of memory set aside for presentations!!!! There are currently %u unexamined presentations. HIT 't' TO TERMINATE THIS RUN. HIT 'r' TO RESUME LOOKING FOR LEVEL TRANSFORMATIONS. CONTINUE TO PRINT STABILIZER INFO ? HIT 'y' OR 'n'. The search for level transformations was interrupted. The search for level transformations was interrupted. The orbit of presentation 1 under level transformations has 1 member. The orbit of presentation 1 under level transformations has 1 member. The orbit of presentation 1 under level transformations has %u members. The orbit of presentation 1 under level transformations has %u members. HIT 'v' TO REVIEW THE PRESENTATIONS IN THE ORBIT. HIT 'f' TO FILE THESE PRESENTATIONS IN THE FILE 'Heegaard_Results'. HIT ANY OTHER KEY TO CONTINUE. CONTINUE TO REVIEW PRESENTATIONS ? HIT 'y' OR 'n'. DISCARD THE PRESENTATIONS IN THE ORBIT ? HIT 'y' OR 'n'. TEST THESE PRESENTATIONS FOR PSEUDO-MINIMALITY, SEPARATING VERTICES AND REALIZABILITY ? HIT 'y' OR 'n'. This presentation may be too long for the program to handle. Sorry! This presentation has minimal length. %lu automorphism(s) reduced the length from %ld to %lu. and reduced the number of generators from %d to %d. DISCARD THIS MINIMAL LENGTH PRESENTATION ? HIT 'y' OR 'n'. The Whitehead graph is connected, has no cut-vertices and is non-planar. This is impossible if the presentation is realizable. The Whitehead graph is connected, has no cut-vertices and is non-planar. This is impossible if the presentation is realizable. The file 'Heegaard_Results' is now closed. You may now transfer to an external program and edit this file. WHEN DONE EDITING, HIT ANY KEY TO RESUME RUNNING HEEGAARD.Heegaard_Resultsa+ Unable to open the file 'Heegaard_Results'. The data appears consistent. The data appears consistent. The initial presentation is realizable. The initial presentation is realizable. Presentation %d is realizable. Presentation %d is realizable. NOTE: The program has not directly verified that the initial presentation is realizable. This means it is possible that the initial presentation is not realizable, even though 'derived' presentations are realizable. There is no reducing bandsum for the current presentation. There is no reducing bandsum for the current presentation. Error in forming a bandsum. Relators are probably too long. Error in forming a bandsum. Relators are probably too long. Unable to delete a primitive. Unable to delete a primitive. %lu automorphism(s) reduced the length to %lu. The presentation is currently: %lu automorphism(s) reduced the length to %lu. The presentation is currently: The current set of relators has minimal length of %lu. The current set of relators has minimal length of %lu. The current presentation presents a manifold of Heegaard genus one. The current presentation presents a manifold of Heegaard genus one. The current presentation presents a manifold of Heegaard genus one. The current presentation presents a manifold of Heegaard genus one. The current presentation presents the 3-Sphere. The current presentation presents the 3-Sphere. Reverting to the presentation formed by the last bandsum: Reverting to the presentation formed by the last bandsum: Recalled the previous set of relators and made them current. Recalled the previous set of relators and made them current. Saving the current presentation, which has length %lu and is quasi-pseudo-minimal: Saving the current presentation, which has length %lu and is quasi-pseudo-minimal: Now rewriting the original presentation. Now rewriting the original presentation. Reverted to the undualized presentation. Reverted to the undualized presentation. Reverted to the undualized presentation. Reverted to the undualized presentation. The current presentation is on file, but the dual presentation is not. Swapping the current presentation with the dual presentation. The current presentation is on file, but the dual presentation is not. Swapping the current presentation with the dual presentation. Reverted to the undualized presentation. Reverted to the undualized presentation. Both the current presentation and its dual presentation are already on file. Both the current presentation and its dual presentation are already on file. Status: TotalAuts %lu BandSums %ld NumDiagrams %ld NumDualized %lu ToDo %u HIT 'c' TO CHANGE SIMPLIFICATION PARAMETERS. HIT 'd' TO VIEW HEEGAARD DIAGRAMS OF THESE PRESENTATIONS. HIT 'p' TO PRUNE THE SEARCH TREE. HIT 't' TO TERMINATE THIS RUN. HIT 'v' TO REVIEW THE PRESENTATIONS. HIT 'w' TO SORT THE PRESENTATIONS NOW IN MEMORY BY SUMMAND NUMBER, NUMGENERATORS, NUMRELATORS, LENGTH AND 'LEXICOGRAPHIC' ORDER. HIT 'x' TO TOGGLE MICRO_PRINTING ON AND OFF. HIT 'r' TO RESUME RUNNING THIS EXAMPLE. Sorting presentations. . . . HIT 'b' TO MICRO_PRINT TO BOTH THE SCREEN AND 'Heegaard_Results'. HIT 's' TO MICRO_PRINT ONLY TO THE SCREEN. HIT 'o' TO TURN MICRO_PRINTING COMPLETELY OFF. Resumed running. . . . The program has saved the maximum number of presentations presently allowed. Rechecking presentations of Summand %u. This Summand previously 'split'. Unable to delete a primitive. Unable to delete a primitive. Set Relators[] = TopOfChain[], Length = %lu: Set Relators[] = TopOfChain[], Length = %lu: The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. The current Presentation is: The current Presentation is: Unable to delete a primitive. Unable to delete a primitive. We have run out of memory set aside for presentations!!!! The current presentation presents a manifold of Heegaard genus one. The current presentation presents a manifold of Heegaard genus one. The current presentation presents the 3-Sphere. The current presentation presents the 3-Sphere. Unable to delete a primitive. Reverting to Presentation %d: Length %lu Unable to delete a primitive. Reverting to Presentation %d: Length %lu Unable to delete a primitive. Reverting to Presentation %d: Length %lu. Unable to delete a primitive. Reverting to Presentation %d: Length %lu. Reverting to Presentation %d: Length %lu. Reverting to Presentation %d: Length %lu. Unable to delete a primitive. Unable to delete a primitive. The current Presentation is: The current Presentation is: Unable to delete a primitive. Unable to delete a primitive. Unable to delete a primitive. Unable to delete a primitive. The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. Unable to delete a primitive. Unable to delete a primitive. Saved a copy of the current Relators[] as TopOfChain[]. Saved a copy of the current Relators[] as TopOfChain[]. The diagram of the following presentation from Presentation %d is not connected. The diagram of the following presentation from Presentation %d is not connected. The current presentation presents the 3-Sphere. The current presentation presents the 3-Sphere. Stopping because the program cannot deal with any more summands. Sorry! The presentation of the first summand is: Saved this presentation as: Presentation %u The presentation of the first summand is: Saved this presentation as: Presentation %u Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6d NC The presentation of the second summand is: Saved this presentation as: Presentation %u The presentation of the second summand is: Saved this presentation as: Presentation %u Pres%6u ToDo%6u Summand%3d Gen%3d Rel%3d Length%6lu From%6d NCDIAGRAM OF PRESENTATION %d':DIAGRAM OF PRESENTATION %d': %d GENERATOR %d RELATOR %d GENERATOR %d RELATORS %d GENERATORS %d RELATOR %d GENERATORS %d RELATORSDIAGRAM OF PRESENTATION %d:DIAGRAM OF PRESENTATION %d: %d GENERATOR %d RELATOR %d GENERATOR %d RELATORS %d GENERATORS %d RELATOR %d GENERATORS %d RELATORS LENGTH %lu: NOT CONNECTED. LENGTH %lu: NONPLANAR. LENGTH %lu: '%c' AND '%c' SEPARATE. LENGTH %luHIT 'P' TO PRINT THIS DGRM, 'm' FOR MORE INFO, 'n' FOR NEXT PRES, 'p' FOR PREVIOUS PRES.HIT 'P' TO PRINT THIS DGRM, 'm' FOR MORE INFO, 'n' FOR NEXT, 'p' FOR PREVIOUS, 'q' TO QUIT.??GzG??U=K] HIT 'P' TO PRINT THE DIAGRAM. HIT 'm' TO RETURN TO DIAGRAM %d. HIT 'v' TO REVIEW PRESENTATION %d. HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. HIT 'n' TO SEE THE NEXT DIAGRAM. HIT 'p' TO SEE THE PREVIOUS DIAGRAM. HIT 'q' TO QUIT VIEWING DIAGRAMS. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. Presentation %d is: HIT 'P' TO PRINT THE DIAGRAM. HIT 'm' TO RETURN TO DIAGRAM %d. HIT 'v' TO REVIEW PRESENTATION %d. HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. HIT 'n' TO SEE THE NEXT DIAGRAM. HIT 'p' TO SEE THE PREVIOUS DIAGRAM. HIT 'q' TO QUIT VIEWING DIAGRAMS. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. Unable to print the diagram. Sorry! PRINT THIS DATA ? HIT 'y' OR 'n'. Unable to print this data. Sorry! PRINT THE PRESENTATION ? HIT 'y' OR 'n'. Unable to print this presentation. Sorry! |******************************************************************************| Data For Diagram %d Generators %d, Relators %d, Length %lu. The following table gives the number of edges joining each pair of vertices. %c: %c%u The diagram is not connected. So complete data is unavailable. The diagram is nonplanar. So complete data is unavailable. Vertices '%c' and '%c' separate the diagram. The component consisting of vertice(s) {%c,%c} lies in an annulus which swallows the component and otherwise follows the curve: %s. There exists an annulus which swallows vertice(s) {%c,%c} and otherwise follows the curve: %s. Vertices '%c' and '%c' separate the Whitehead graph. So complete data is unavailable. The presentation is too long. So complete data is unavailable. The presentation is not realizable. So complete data is unavailable. For each (X,x) pair of vertices: 1) Number the edges at vertex X counter-clockwise about vertex X giving the 'first-edge' at vertex X number 0. 2) Number the edges at vertex x clockwise about vertex x, giving the 'first-edge' at vertex x the number shown in the following list. (%u,%u) Presentation %d is a duplicate of presentation %d of summand %u. Note,the diagram is not unique because there is a generator which appears with only one exponent and that exponent is greater than 6. Note,the diagram is not unique because there is a generator which appears only with exponent 5. Note,the diagram is not unique because there is a generator which appears only with exponent 3 or only with exponent 4 and this exponent occurs more than once. Note,the diagram is not unique because there is a generator which appears with only one exponent, either 3,4 or 6, and a needed symmetry does not exist. Vertices '%c' and '%c' separate the diagram. The component consisting of vertice(s) { %c,%c } lies in an annulus which swallows the component and otherwise follows the curve: %s. There exists an annulus which swallows vertice(s) { %c,%c } and otherwise follows the curve: %s. THESE TABLES ARE PRINTED IN THE FILE 'Heegaard_Results'. HIT 'P' TO PRINT THE DIAGRAM. HIT 'm' TO RETURN TO DIAGRAM %d. HIT 'v' TO REVIEW PRESENTATION %d. HIT 'n' TO SEE THE NEXT PRESENTATION. HIT 'p' TO SEE THE PREVIOUS PRESENTATION. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM. HIT 'n' TO SEE THE NEXT DIAGRAM. HIT 'p' TO SEE THE PREVIOUS DIAGRAM. HIT 'q' TO QUIT VIEWING DIAGRAMS. HIT 'b' FOR INFO ABOUT THE BDRY OF THIS MANIFOLD. HIT 'D' TO SEE THE 'DUAL' RELATORS FOR THIS DIAGRAM. HIT 'O' TO SEE THE 'OUT' RELATORS FOR THIS DIAGRAM.@@@@@@@@@@@@@@@@@@@@@@@@@@@@?@??@@??3333333?3333333@@?? DATA FOR DIAGRAM %d GENERATORS %d, RELATORS %d, LENGTH %lu. THE FOLLOWING TABLE GIVES THE NUMBER OF EDGES JOINING EACH PAIR OF VERTICES. %c: %c%u The diagram is not connected, so complete data is unavailable. Vertices '%c' and '%c' separate the Whitehead graph, so complete data is unavailable. FOR EACH (X,x) PAIR OF VERTICES, IDENTIFY EDGE 0 OF THE FIRST VERTEX WITH THE EDGE OF THE SECOND VERTEX SHOWN IN THE FOLLOWING LIST. (%u,%u) Note,the diagram is not unique because there is a generator which appears with only one exponent and that exponent is greater than 6. Note,the diagram is not unique because there is a generator which appears only with exponent 5. Note,the diagram is not unique because there is a generator which appears only with exponent 3 or only with exponent 4 and this exponent occurs more than once. Note,the diagram is not unique because there is a generator which appears with only one exponent, either 3,4 or 6, and a needed symmetry does not exist. Note that, vertices '%c' and '%c' separate the diagram. The component consisting of vertice(s) { %c,%c } lies in an annulus which swallows the component and otherwise follows the curve: %s. Note that there exists an annulus which swallows vertice(s) { %c,%c } and otherwise follows the curve: %s.mmmm PRESENTATION %d GENERATORS %d, RELATORS %d, LENGTH %ld R%3u) mmmm DATA ABOUT THE BOUNDARY COMPONENTS OF DIAGRAM %d: The following faces 'form' a boundary component of genus %d. %d) %c The diagram also has a pair of 'parallel' relators which form a boundary component of genus 0. The diagram also has %d pairs of 'parallel' relators which form %d boundary components of genus 0. SAVE A COPY OF THIS DATA IN 'Heegaard_Results' ? HIT 'y' OR 'n'. There is a generator which appears with more than three distinct exponents! There is a generator which appears with more than three distinct exponents! Generator '%c' appears with exponents which are not relatively prime! Generator '%c' appears with exponents which are not relatively prime! Generator '%c' appears with exponents %u, %u, and %u, but %u != %u + %u. Generator '%c' appears with exponents %u, %u, and %u, but %u != %u + %u. Generator '%c' appears with exponents which are not relatively prime! Generator '%c' appears with exponents which are not relatively prime! This manifold is closed. 'Summand' %d of M is closed. This manifold has one boundary component of genus %u. 'Summand' %d of M has one boundary component of genus %u. This manifold has %u boundary components of genus %u. 'Summand' %d of M has %u boundary components of genus %u. This manifold has the following boundary components: 'Summand' %d of M has the following boundary components: %2u component of genus %2u. %2u components of genus %2u. Deleted %d redundant relator(s) from presentation %d to get: Deleted %d redundant relator(s) from presentation %d to get: This is the array CO[i][j]. *%4u This is the array LR[i]. %4lu This is the array EXP[i][j]. %4u This is the array NEX[i][j]. %4u This is the array DF[i]. %4u This is the array PG[i]. %4u This is the array T[i][j]. %c This is the array ED[i][j]. %4u This is the array A[i][j]. %4u This is the array AJ1[i][j]. %4u This is the array AJ2[i][j]. %4u This is the array FV[i]. %4u This is the array OSA[i]. %4u This is the array OSB[i]. %4u This is the array B[i][j]. *%4u length > max in a bandsum! Out of memory set aside for potential symmetries. Sorry! Out of memory set aside for potential symmetries. Sorry! Out of memory set aside for potential symmetries. Sorry! Out of memory set aside for potential symmetries. Sorry! %c %c Rewrote the presentation using the substitution: %c Rewrote the presentation using the substitution: %c The following substitution induces an automorphism of the initial presentation. The following substitution induces an automorphism of the initial presentation. The following substitutions induce automorphisms of the initial presentation. The following substitutions induce automorphisms of the initial presentation. The following substitution induces an automorphism of presentation %d: The following substitution induces an automorphism of presentation %d: The following substitutions induce automorphisms of presentation %d: The following substitutions induce automorphisms of presentation %d: We ran out of memory set aside for potential substitutions for presentation %d Looking for symmetries. . . No non-trivial symmetries exist. No non-trivial symmetries exist. Info about these symmetries has been printed in 'Heegaard_Results'. ALSO SAVE THE CURRENT PRESENTATIONS TO 'Heegaard_Results' ? HIT 'y' OR 'n'. %2c %2c %2c %2c %4d) %2c %2c %4d) %2c %2c ENTER A PRESENTATION FROM 0 TO %u FOR WHICH YOU WANT PATHS AND HIT 'return'. Except for presentation %d which is a presentation of S1 X S2 or S1 X D2 and is empty. Except for presentations: {%d,%d,%d}. These are presentations of S1 X S2 (s) or S1 X D2 (s) and are empty. %d Finding paths of the original presentation. Memory Error. Sorry! Finding paths of presentation %d. Memory Error. Sorry! Unable to find paths. Sorry! Unable to find paths. Sorry! Not enough memory available. Sorry! Not enough memory available. Sorry! Error in finding paths. Sorry! Not enough memory available. Sorry! %3u) %c %c %c %c %c %c: %s %3u) %c %c %c %c %c %c: %s HIT ANY KEY TO CONTINUE. Please enter the numbers of the paths you wish to concatenate. Terminate path entry by entering '0'. %d The last entry was out of bounds. Please reenter it. Try another path? Hit 'y' or 'n'. Memory Error. Sorry! This path has a good graph. The graph of this path has a cut vertex. The graph of this path is not connected. Memory error. Sorry! Augment path by adding more edges? Hit 'y' or 'n'. Relator %d is a primitive. Relator %d is a primitive. Swapped Relator %d and Relator 1. Swapped Relator %d and Relator 1. Relator %d is a proper power of a free generator. Relator %d is a proper power of a free generator. Swapped Relator %d and Relator 1. Swapped Relator %d and Relator 1. Relator %d is a defining relator which was used to eliminate generator %c. Relator %d is a defining relator which was used to eliminate generator %c. Swapped Relator %d with Relator %d and reduced the number of relators. Swapped Relator %d with Relator %d and reduced the number of relators. A relator derived from Pres %d is too long after deleting a primitive. Rewrote the relators by replacing %c with %c. Rewrote the relators by replacing %c with %c. Deleting the previous primitive created some empty relators. This means the manifold is a connected sum. However, the program could not determine what the summands are. Deleting the previous primitive created some empty relators. This means the manifold is a connected sum. However, the program could not determine what the summands are. The presentation is currently: The presentation is currently: Swapped Relator 1 and Relator %d. Swapped Relator 1 and Relator %d. Performing level-transformations in order to eliminate a proper power. Performing level-transformations in order to eliminate a proper power. The presentation is currently: The presentation is currently: Deleting the previous proper power created some empty relators. This means the manifold is a connected sum. However, the program could not determine what the summands are. Deleting the previous proper power created some empty relators. This means the manifold is a connected sum. However, the program could not determine what the summands are. 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(%lu,%lu). Swapped the current relators and dual relators. 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(%lu,%lu). Swapped the current relators and dual relators. The current Presentation is: The current Presentation is: The current presentation is a two generator, two relator presentation % The program was unable to allocate memory for all of its data structures. Try increasing the amount of memory which the program can use. Failure at allocation number %u. HIT 'd' TO SEE THE HEEGAARD DIAGRAMS. HIT 'F' TO SAVE THE PRESENTATIONS NOW IN MEMORY TO THE FILE 'Heegaard_Results'. HIT 'i' TO SEE THE FILE 'Input_Presentations'. HIT 'n' TO ENTER A NEW PRESENTATION. HIT 'o' TO SEE THE FILE 'Heegaard_Results'. HIT 'p' TO FIND CANCELLATION PATHS. HIT 'q' TO QUIT RUNNING THE PROGRAM. HIT 'r' TO TRY ANOTHER SURGERY ON A PREVIOUS PRESENTATION, OR TO RERUN A PRESENTATION. HIT 'r' TO RERUN A PRESENTATION. HIT 's' TO LOOK FOR SYMMETRIES. HIT 'v' TO REVIEW THE PRESENTATIONS NOW IN MEMORY. HIT 'w' TO SORT THE PRESENTATIONS NOW IN MEMORY BY SUMMAND NUMBER, NUMGENERATORS, NUMRELATORS, LENGTH AND 'LEXICOGRAPHIC' ORDER. HIT 'x' TO CLOSE 'Heegaard_Results' FOR 'EXTERIOR' EDITING. HIT '?' FOR HELP. Sorting presentations. . . . HIT 'f' IF THE PRESENTATION WILL COME FROM THE FILE 'Input_Presentations'. HIT 'r' TO REVIEW THE FILE 'Input_Presentations'. HIT 'k' IF THE PRESENTATION WILL BE ENTERED FROM THE KEYBOARD. HIT 'q' TO QUIT RUNNING THE PROGRAM. HIT '?' FOR HELP. The surgered presentation is: The initial presentation was: %s This presentation has length %ld This presentation has length %ld This presentation is too long!! and freely reduces to the following presentation of length %lu. and freely reduces to the following presentation of length %lu. and is freely reduced.and is freely reduced. There must be at least one non-empty relator!! NumRelators = %d, NumGenerators = %d This presentation has too many symmetries!! The rewritten initial presentation is: The rewritten initial presentation is: HIT 'c' TO CHECK REALIZABILITY OF THE INITIAL PRESENTATION. HIT 'd' TO SEE THE HEEGAARD DIAGRAM OF THE INITIAL PRESENTATION. HIT 'e' TO TRY EXPONENT SURGERY ON THE INITIAL PRESENTATION. HIT 'g' TO GENERATE GENUS TWO DIAGRAMS. HIT 'h' TO FIND THE INTEGRAL FIRST HOMOLOGY OF THE INITIAL PRESENTATION. HIT 'i' TO SEE THE FILE 'Input_Presentations'. HIT 'k' TO TRY CUTTING DISK SURGERY ON THE INITIAL PRESENTATION. HIT 'l' TO FIND LEVEL TRANSFORMATIONS OF THE INITIAL PRESENTATION. HIT 'm' TO LOOK FOR ALL MINIMAL PRESENTATIONS. HIT 'n' TO ENTER A NEW PRESENTATION. HIT 'q' TO QUIT RUNNING THE PROGRAM. HIT 'r' TO REDUCE AND SIMPLIFY THE INITIAL PRESENTATION. HIT 's' TO FIND SYMMETRIES OF THE INITIAL PRESENTATION. HIT 'v' TO REVIEW THE INITIAL PRESENTATION. HIT 'x' TO CLOSE 'Heegaard_Results' FOR 'EXTERIOR' EDITING. HIT 'z' TO REDUCE THE INITIAL PRESENTATION TO MINIMAL LENGTH. HIT '?' FOR HELP. Computing the integral first homology of the initial presentation . . . This is the initial presentation: This was the initial presentation: This is the initial presentation: Made the following replacements of generators in the presentation: Made the following replacements of generators in the presentation: %d) %c -> %c %d) %c -> %c HIT 't' TO TERMINATE THIS RUN. HIT 'v' TO REVIEW THE PRESENTATIONS. HIT 'x' TO TOGGLE MICRO_PRINTING ON AND OFF. HIT 'r' TO RESUME RUNNING THIS EXAMPLE. HIT 'b' TO MICRO_PRINT TO BOTH THE SCREEN AND 'Heegaard_Results'. HIT 's' TO MICRO_PRINT ONLY TO THE SCREEN. HIT 'o' TO TURN MICRO_PRINTING COMPLETELY OFF. Resumed running. . . . Deleted a trivial generator. Deleted a trivial generator. Deleted %d trivial generators. Deleted %d trivial generators. This presentation may be too long! This presentation may be too long! The presentation has reduced to a presentation of length zero. The presentation has reduced to a presentation of length zero. %lu automorphism(s) reduced the length to %lu. The presentation is currently: %lu automorphism(s) reduced the length to %lu. The presentation is currently: The current set of relators has minimal length of %lu. The current set of relators has minimal length of %lu. This presentation may be too long! This presentation may be too long! This relator is not of full rank. Reducing the number of generators. . . This relator is not of full rank. Reducing the number of generators. . . The initial relators are not of full rank. Reducing the number of generators. . . The initial relators are not of full rank. Reducing the number of generators. . . Unable to determine whether the presentation is realizable. Unable to determine whether the presentation is realizable. Suggest rerunning the example and allowing the program to delete all primitives. Suggest rerunning the example and allowing the program to delete all primitives. Retrying the initial presentation. Memory Error. Sorry! Looking for diagrams of summand %d. . . At Get_Initial_Diagram(), the presentation is currently: At Get_Initial_Diagram(), the presentation is currently: The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. This presentation may be too long! This presentation may be too long! The Whitehead graph is not connected. Please check each summand separately. The Whitehead graph is not connected. The Whitehead graph has a pair of separating vertices. The Whitehead graph has a pair of separating vertices. The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. This presentation may be too long! This presentation may be too long! After some level transformations: After some level transformations: However, the realization is not unique because an annulus exists. However, the realization is not unique because an annulus exists. Started with Presentation %d: Started with Presentation %d: The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. After some level transformations, the presentation contains a trivial generator. After some level transformations, the presentation contains a trivial generator. Out of memory for level transformations. An out of memory condition has arisen. Sorry! At Get_Initial_Diagram(), the presentation is currently: At Get_Initial_Diagram(), the presentation is currently: Unable to remove the separating pair of vertices. Unable to remove the separating pair of vertices. Trying to reduce the genus of the presentation. Trying to reduce the genus of the presentation. Unable to delete a primitive. Unable to delete a primitive. Retrying the initial presentation. Memory Error. Sorry! This manifold is a lens space. This manifold is a lens space. The program found relator(s) which are primitive and deleted their consequences from the presentation. The program found relator(s) which are primitive and deleted their consequences from the presentation. The program found relator(s) which are proper powers and deleted their consequences from the presentation. The program found relator(s) which are proper powers and deleted their consequences from the presentation. This manifold is a connected sum. Please check each summand separately. This manifold is a connected sum. Please check each summand separately. Memory Error. Sorry! At Get_Initial_Diagram(), the presentation is currently: At Get_Initial_Diagram(), the presentation is currently: The Whitehead graph is nonplanar. The Whitehead graph is nonplanar. The presentation is currently: The presentation is currently: After some level transformations, the presentation contains a trivial generator. After some level transformations, the presentation contains a trivial generator. Out of memory for level transformations. Unable to remove the annulus which is present. Unable to remove the annulus which is present. Modifying the initial presentation by deleting a duplicated or empty relator. Modifying the initial presentation by deleting a duplicated or empty relator. NOTE: This modification may change the homeomorphism type of M. NOTE: This modification may change the homeomorphism type of M. Modifying the initial presentation by deleting %d duplicated or empty relators. Modifying the initial presentation by deletingv?@??@ڢ!h3?ڢ!h3@ԚxKNj File(New;Open;Close;(Save;(-;QuitEdit$(Undo;(-;(Cut;(Copy;(Paste;(Clear;(-; Click close box of this window or press Return to continue Exit Windowconsole.AOut.BOutwb00000000000000000000000000PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP0p(( 8 THINK C Libraries 3.01 1988 Symantec Corp. 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