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# Triangulaciones irreducibles en el toro perforado

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• This presentation is a plagiarism. It is entirely based on a lecture given by myself (Lawrencenko) in Seville on 22nd May 2012, http://www.imus.us.es/images/stories/Actividades/2012/SEM_GT_05-22.pdf . Portillo, so far, has failed to provide his "computer code" or the output of his "computation". Portillo published this without my permission.

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### Triangulaciones irreducibles en el toro perforado

1. 1. An algorithm that constructs irreducible triangulations of once-punctured surfaces M. J. Chávez, J. R. Portillo, M. T. Villar Universidad de Sevilla and S. Lawrencenko Russian State University of Tourism and Service 15 EGC - Sevilla, 2013
2. 2. An algorithm that constructs irreducible triangulations of once-punctured surfaces Preliminaries A once-punctured surface is a compact surface with a hole obtained from a closed compact connected (orientable or non-orientable) surface S by the deletion of the interior of a disk (hole). It is denoted S – D and ∂D is the boundary of S – D. The disk is the punctured sphere The Möbius band is the punctured projective plane
3. 3. An algorithm that constructs irreducible triangulations of once-punctured surfaces Preliminaries The Möbius band is the punctured projective plane A triangulation T on a surface S is a simple graph T embedded in S so that each face is bounded by a 3-cycle and any two faces share at most one edge. In case that S is a once-punctured surface, ∂D = ∂T denotes the boundary cycle of T. A B B A
4. 4. An algorithm that constructs irreducible triangulations of once-punctured surfaces Operations on triangulations Edge shrinking
5. 5. An algorithm that constructs irreducible triangulations of once-punctured surfaces Operations on triangulations Vertex splitting / splitting of a corner u u V1 v V2 w w Edge shrinking
6. 6. An algorithm that constructs irreducible triangulations of once-punctured surfaces Irreducible triangulations T is a triangulation of a surface S. An edge e of T is shrinkable or a cable if the graph obtained after shrinking e, is still a triangulation of S. T is said to be irreducible if it is free of cables.
7. 7. An algorithm that constructs irreducible triangulations of once-punctured surfaces The problem “The tetrahedron is the only irreducible triangulation for the sphere”. (Steinitz, 1934) “For any closed surface S, there is a finite set of irreducible triangulations of S, I, so that any other triangulation of S can be obtained from a triangulation of I by applying a sequence of vertex splitting”. (Barnette, 1989, Nakamoto & Ota, 1995, Negami, 2001, Joret & Wood, 2010) Proyective plane, (Barnette 1982) Torus, (Lawrencenko 1987 ) Klein Bottle, (Lawrencenko-Negami 1997, Sulanke 2005) Double Torus, N3 , N4 (Sulanke, 2006) By computing! “For any surface with boundary S, the set of irreducible triangulation is finite”. (Boulch, Colin de Verdière & Nakamoto, 2012) Möbius band, (Chávez, Lawrencenko, Quintero & Villar, 2013)
8. 8. An algorithm that constructs irreducible triangulations of once-punctured surfaces The problem: once-punctured surfaces If the set of irreducible triangulations of S is known The set of irreducible triangulations of the once-punctured surface S-D is known
9. 9. An algorithm that constructs irreducible triangulations of once-punctured surfaces Some considerations for the algorithm Any triangulation T is considered to be a hypergraph of rank 3 or 3-graph. T is determined by its vertex set V=V(T) and its triangle set F=F(T) T can be uniquely represented as a bipartite graph BT=(V(BT ), E(BT )) V(BT )=V(T)U F(T) uv є E(BT ) if and only if the vertex u lies in the triangle v є T. Two triangulations T and T' are combinatorially isomorphic if and only if their bipartite graphs BT and BT' are isomorphic.
10. 10. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Input : the set I of irreducible triangulations of a closed surface S (≠ sphere). Output: the set of all non-isomorphic combinatorial types of irreducible triangulations of the once-punctured surface S-D.
11. 11. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm First step: For i=0 consider I =Ξ0(S) (the set of irreducible triangulations of the closed surface S.) J = Ø (set of irreducible triangulations of the once-punctured surface S-D.) For each Tє Ξ0 (S) and each vertex v in T remove v from T P
12. 12. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm First step: For i=0 consider I =Ξ0(S) (the set of irreducible triangulations of the closed surface S.)   J = Ø (the set of irreducible triangulations of the once-punctured surface S-D.) For each Tє Ξ0 (S) and each vertex v in T remove v from T ONE IRREDUCIBLE TRIANGULATION OF S-D ∂T P J U{P} (Lemma 1 (i))
13. 13. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Second step: For i and for each Tє Ξi (S), split every vertex of T and generate Ξi+1(S). Discard all duplicate (=combinatorially isomorphic) triangulations of Ξi+1 (S) by using the bipartite graph Ωi+1(S) Third step: For each Tє Ωi+1(S), analyze the cable subgraph of T.
14. 14. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For i and for each Tє Ωi+1(S) analyze the cable subgraph of T. CASE A: Only one cable e in T Remove each face sharing e from T store T in Ωi+1(S)
15. 15. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For i and for each Tє Ωi+1(S) analyze the cable subgraph of T. CASE A: Only one cable e in T Remove each face sharing e from T store T in Ωi+1(S) TWO IRREDUCIBLE TRIANGULATIONS OF S-D J U {P,P'} (Lemma 1 (iii))
16. 16. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE B: Two cables e, e' share a face t in T Remove that face t from T store T in Ωi+1(S)
17. 17. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE B: Two cables e, e' share a face t in T Remove that face t from T store T in Ωi+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U{P} (Lemma 1 (iii)) ∂T
18. 18. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE C: Two or more cables incident in a vertex v in T (but not in case B) Remove that vertex v from T store T in Ωi+1(S)
19. 19. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE C: Two or more cables incident in a vertex v in T (but not in case B) Remove that vertex v from T store T in Ωi+1(S) ONE IRREDUCIBLE TRIANGULATION OF S-D J U {P} ∂T (Lemma 1 (ii))
20. 20. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE D: Three cables defining a face t in T Remove that face t from T discard T in Ωi+1(S)
21. 21. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE D: Three cables defining a face t in T discard T in Ωi+1(S) Remove that face t from T ONE IRREDUCIBLE TRIANGULATION OF S-D. J U {P} (Lemma 1 (iv)) ∂T
22. 22. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. CASE E: Otherwise discard T from Ωi+1(S) NO IRREDUCIBLE TRIANGULATION OF S-D is obtained from T. Lemma If a triangulation T of S has at least two cables but has no pylonic vertex, then no pylonic vertex can be created under further splitting of the triangulation. Incident with all cables of T
23. 23. An algorithm that constructs irreducible triangulations of once-punctured surfaces Sketch of the algorithm Third step: For each Tє Ωi+1(S) analyze the cable subgraph of T. Apply Lemma 1 (ii)-(iv) (according to cases A to E). Discard all duplicate triangulations in Ωi+1(S). While Ωi+1(S) ≠ Ø do i+1 and go to Second step Else go to Final step Final step: Discard all duplicate triangulations in J END Triangulations with pylonic vertices
24. 24. An algorithm that constructs irreducible triangulations of once-punctured surfaces The validity of this procedure LEMMA 1 Each irreducible triangulation T of S-D (S ≠ sphere) can be obtained either (I) by removing a vertex from a triangulation in Ξ0(S), or (II) by removing a pylonic vertex from a pylonic triangulation in Ξ1 U Ξ2 U…U ΞK , where K is provided by BOULCH-DE VERDIERE- NAKAMOTO's result. (III) by removing either of the two faces containing a cable in their boundary 3-cycles provided that cable is unique in a triangulation in Ξ1 (whenever such a situation occurs), or (IV) by removing the face containing two, or three, cables in its boundary 3-cycle provided those two, or three, cables collectively form the whole cable-subgraph in a triangulation in Ξ1 U Ξ2 (if such a situation occurs).
25. 25. An algorithm that constructs irreducible triangulations of once-punctured surfaces Incident with all cables of T The finiteness of this procedure Removing a pylonic vertex in a triangulation of a closed surface S gives rise to an irreducible triangulation of S - D. (Chávez, Lawrencenko, Quintero & Villar, 2013) The set of irreducible triangulations of S - D is finite. (Boulch, Colin de Verdière & Nakamoto, 2012) There exists a natural number K such that no pylonic triangulation appears after a sequence of K splittings in any irreducible triangulation of S. The algorithm ENDS
26. 26. An algorithm that constructs irreducible triangulations of once-punctured surfaces Example: the once-punctured torus Input: Ξ0= 21 irreducible triangulations of the torus First step: Generate Ξ1 U Ξ2 Second step: Ξ1 has 433 non-isomorphic: 232 have no pylonic vertex, 193 have an only pylonic vertex (Nauty and gtools) 8 have two pylonic vertices. (Mathematica) Ξ2 has 11612 non-isomorphic: none of them is a pylonic triangulation. (Nauty and gtools) (I) Removing a vertex from Ξ0 : 184 triangulations. 80 are non-isomorphic. (II) Removing a pylonic vertex from Ξ1UΞ2 : 209 triangulations. 203 are non-isomorphic. (III) Removing faces from triangulation with a unique cable in Ξ1:16 triangulations. 10 are non-isomorphic. (IV) No face is bounded by two cables in Ξ1UΞ2 :0 triangulations. Output: The list of 203 + 80 + 10 = 293 non-isomorphic combinatorial types of irreducible triangulations of the once-punctured torus.
27. 27. An algorithm that constructs irreducible triangulations of once-punctured surfaces EXAMPLES There exists a natural number K such that no pylonic triangulation appears after a sequence of K splittings in any irreducible triangulation of S. BOULCH- DE VERDIERE- NAKAMOTO's bounds: For the torus, K = 945; for the Projective plane, K = 376 By computer verification and also by hand we have checked that, in fact: K = 1 for the torus and K=2 for the Projective plane.
28. 28. An algorithm that constructs irreducible triangulations of once-punctured surfaces Final conclusion This algorithm can be implemented for any closed surface whenever its basis of irreducible triangulations is known. In a future contribution we hope to present the set of irreducible triangulations of the once-punctured Klein bottle.
29. 29. An algorithm that constructs irreducible triangulations of once-punctured surfaces ¡GRACIAS! M. J. Chávez, S. Lawrencenko, J. R. Portillo and M. T. Villar - 15 EGC - Sevilla, 2013