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Partial duality of hypermaps Sergei Chmutov Ohio State University, - PowerPoint PPT Presentation

Partial duality of hypermaps Sergei Chmutov Ohio State University, Mansfield Conference Legacy of Vladimir Arnold , Fields Institute, Toronto. Joint with Fabien Vignes-Tourneret arXiv:1409.0632 [math.CO] Tuesday, November 25, 2014


  1. Partial duality of hypermaps Sergei Chmutov Ohio State University, Mansfield Conference Legacy of Vladimir Arnold , Fields Institute, Toronto. Joint with Fabien Vignes-Tourneret arXiv:1409.0632 [math.CO] Tuesday, November 25, 2014 9:00–9:30am Sergei Chmutov Partial duality of hypermaps

  2. Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps

  3. Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps

  4. Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps

  5. Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps

  6. Hypermaps Sergei Chmutov Partial duality of hypermaps

  7. Hypermaps Sergei Chmutov Partial duality of hypermaps

  8. Hypermaps Sergei Chmutov Partial duality of hypermaps

  9. Hypermaps Sergei Chmutov Partial duality of hypermaps

  10. Hypermaps Sergei Chmutov Partial duality of hypermaps

  11. τ -model for hypermaps a flag a face a vertex a (hyper) edge Sergei Chmutov Partial duality of hypermaps

  12. τ -model for hypermaps a flag a face a vertex a (hyper) edge f v’ v e τ 0 (v,e,f) (v’,e,f) Sergei Chmutov Partial duality of hypermaps

  13. τ -model for hypermaps a flag a face a vertex a (hyper) edge e’ f f v v’ v e e τ τ 1 0 (v,e,f) (v,e’,f) (v,e,f) (v’,e,f) Sergei Chmutov Partial duality of hypermaps

  14. τ -model for hypermaps a flag a face a vertex a (hyper) edge e’ f v f e f v f’ v’ v e e τ τ τ 1 2 0 (v,e,f) (v,e’,f) (v,e,f) (v,e,f’) (v,e,f) (v’,e,f) Sergei Chmutov Partial duality of hypermaps

  15. τ -model. Example. 3 10 12 2 1 11 6 7 5 9 8 4 Sergei Chmutov Partial duality of hypermaps

  16. τ -model. Example. 3 10 12 2 1 11 6 7 5 9 8 4 τ 0 = ( 1 , 11 )( 2 , 12 )( 3 , 10 )( 4 , 8 )( 5 , 9 )( 6 , 7 ) Sergei Chmutov Partial duality of hypermaps

  17. τ -model. Example. 3 10 12 2 1 11 6 7 5 9 8 4 τ 0 = ( 1 , 11 )( 2 , 12 )( 3 , 10 )( 4 , 8 )( 5 , 9 )( 6 , 7 ) τ 1 = ( 1 , 2 )( 3 , 4 )( 5 , 6 )( 7 , 9 )( 8 , 10 )( 11 , 12 ) Sergei Chmutov Partial duality of hypermaps

  18. τ -model. Example. 3 10 12 2 1 11 6 7 5 9 8 4 τ 0 = ( 1 , 11 )( 2 , 12 )( 3 , 10 )( 4 , 8 )( 5 , 9 )( 6 , 7 ) τ 1 = ( 1 , 2 )( 3 , 4 )( 5 , 6 )( 7 , 9 )( 8 , 10 )( 11 , 12 ) τ 2 = ( 1 , 6 )( 2 , 3 )( 4 , 5 )( 7 , 11 )( 8 , 9 )( 10 , 12 ) Sergei Chmutov Partial duality of hypermaps

  19. σ -model for oriented hypermaps v σ V Sergei Chmutov Partial duality of hypermaps

  20. σ -model for oriented hypermaps v e σ σ V E Sergei Chmutov Partial duality of hypermaps

  21. σ -model for oriented hypermaps v f e σ σ σ V E F Sergei Chmutov Partial duality of hypermaps

  22. σ -model for oriented hypermaps v f e σ σ σ V E F σ E σ V σ F σ E σ V = 1 : σ F Sergei Chmutov Partial duality of hypermaps

  23. σ -model. Example. 3 12 1 7 5 8 Sergei Chmutov Partial duality of hypermaps

  24. σ -model. Example. 3 12 1 7 5 8 σ V = ( 1 , 3 , 5 )( 7 , 8 , 12 ) = τ 2 τ 1 | { 1 , 3 , 5 , 7 , 8 , 12 } Sergei Chmutov Partial duality of hypermaps

  25. σ -model. Example. 3 12 1 7 5 8 σ V = ( 1 , 3 , 5 )( 7 , 8 , 12 ) = τ 2 τ 1 | { 1 , 3 , 5 , 7 , 8 , 12 } σ E = ( 1 , 7 )( 3 , 12 )( 5 , 8 ) = τ 0 τ 2 | { 1 , 3 , 5 , 7 , 8 , 12 } Sergei Chmutov Partial duality of hypermaps

  26. σ -model. Example. 3 12 1 7 5 8 σ V = ( 1 , 3 , 5 )( 7 , 8 , 12 ) = τ 2 τ 1 | { 1 , 3 , 5 , 7 , 8 , 12 } σ E = ( 1 , 7 )( 3 , 12 )( 5 , 8 ) = τ 0 τ 2 | { 1 , 3 , 5 , 7 , 8 , 12 } σ F = ( 1 , 12 )( 3 , 8 )( 5 , 7 ) = τ 1 τ 0 | { 1 , 3 , 5 , 7 , 8 , 12 } Sergei Chmutov Partial duality of hypermaps

  27. Duality for graphs G Sergei Chmutov Partial duality of hypermaps

  28. Duality for graphs G Sergei Chmutov Partial duality of hypermaps

  29. Duality for graphs G Sergei Chmutov Partial duality of hypermaps

  30. Duality for graphs G ∗ = G { 1 , 2 , 3 , 4 , 5 , 6 } G Sergei Chmutov Partial duality of hypermaps

  31. Partial duality for graphs G Sergei Chmutov Partial duality of hypermaps

  32. Partial duality for graphs G 1 3 6 4 5 2 Sergei Chmutov Partial duality of hypermaps

  33. Partial duality for graphs G { 1 , 2 , 3 , 4 , 5 } = ??? G 1 3 6 4 5 2 Sergei Chmutov Partial duality of hypermaps

  34. Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps

  35. Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps

  36. Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps

  37. Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps

  38. Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps

  39. Partial duality for graphs (continuation) R { 1 , 2 , 3 , 4 , 5 } Sergei Chmutov Partial duality of hypermaps

  40. Partial duality for hypermaps Let S be a subset of the vertex-cells of G . Sergei Chmutov Partial duality of hypermaps

  41. Partial duality for hypermaps Let S be a subset of the vertex-cells of G . Choose a different type of cells, say hyperedges. Sergei Chmutov Partial duality of hypermaps

  42. Partial duality for hypermaps Let S be a subset of the vertex-cells of G . Choose a different type of cells, say hyperedges. Step 1. ∂ F is the boundary a surface F which is the union of the cells from S and all hyperedge-cells. Sergei Chmutov Partial duality of hypermaps

  43. Partial duality for hypermaps Let S be a subset of the vertex-cells of G . Choose a different type of cells, say hyperedges. Step 1. ∂ F is the boundary a surface F which is the union of the cells from S and all hyperedge-cells. Step 2. Glue in a disk to each connected component of ∂ F . These will be the hyperedge-cells for G S . Sergei Chmutov Partial duality of hypermaps

  44. Partial duality for hypermaps (continuation) Step 3. Gluing the vertex-cells. Sergei Chmutov Partial duality of hypermaps

  45. Partial duality for hypermaps (continuation) Step 3. Gluing the vertex-cells. Sergei Chmutov Partial duality of hypermaps

  46. Partial duality for hypermaps (continuation) Step 4. Forming the partial dual hypermap G S . Sergei Chmutov Partial duality of hypermaps

  47. Partial duality for hypermaps (continuation) Step 4. Forming the partial dual hypermap G S . 10 1 2 12 12 3 11 1 1 12 3 8 7 8 5 4 7 5 9 5 8 6 Sergei Chmutov Partial duality of hypermaps

  48. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. Sergei Chmutov Partial duality of hypermaps

  49. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. G S � S = G . � (b) Sergei Chmutov Partial duality of hypermaps

  50. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. G S � S = G . � (b) (c) There is a bijection between the cells of type S in G and the cells of the same type in G S . This bijection preserves the valency of cells. The number of cell of other types may change. Sergei Chmutov Partial duality of hypermaps

  51. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. G S � S = G . � (b) (c) There is a bijection between the cells of type S in G and the cells of the same type in G S . This bijection preserves the valency of cells. The number of cell of other types may change. (d) Is s �∈ S but has the same type as the cells of S , then G S ∪{ s } = G S � { s } . � Sergei Chmutov Partial duality of hypermaps

  52. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. G S � S = G . � (b) (c) There is a bijection between the cells of type S in G and the cells of the same type in G S . This bijection preserves the valency of cells. The number of cell of other types may change. (d) Is s �∈ S but has the same type as the cells of S , then G S ∪{ s } = G S � { s } . � G S � S ′ = G ∆( S , S ′ ) , where ∆( S , S ′ ) := ( S ∪ S ′ ) \ ( S ∩ S ′ ) is � (e) the symmetric difference of sets. Sergei Chmutov Partial duality of hypermaps

  53. Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. G S � S = G . � (b) (c) There is a bijection between the cells of type S in G and the cells of the same type in G S . This bijection preserves the valency of cells. The number of cell of other types may change. (d) Is s �∈ S but has the same type as the cells of S , then G S ∪{ s } = G S � { s } . � G S � S ′ = G ∆( S , S ′ ) , where ∆( S , S ′ ) := ( S ∪ S ′ ) \ ( S ∩ S ′ ) is � (e) the symmetric difference of sets. (f) The partial duality preserves orientability of hypermaps. Sergei Chmutov Partial duality of hypermaps

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