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
Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps
Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps
Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps
Maps (Graphs on surfaces) Sergei Chmutov Partial duality of hypermaps
Hypermaps Sergei Chmutov Partial duality of hypermaps
Hypermaps Sergei Chmutov Partial duality of hypermaps
Hypermaps Sergei Chmutov Partial duality of hypermaps
Hypermaps Sergei Chmutov Partial duality of hypermaps
Hypermaps Sergei Chmutov Partial duality of hypermaps
τ -model for hypermaps a flag a face a vertex a (hyper) edge Sergei Chmutov Partial duality of hypermaps
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
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
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
τ -model. Example. 3 10 12 2 1 11 6 7 5 9 8 4 Sergei Chmutov Partial duality of hypermaps
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
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
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
σ -model for oriented hypermaps v σ V Sergei Chmutov Partial duality of hypermaps
σ -model for oriented hypermaps v e σ σ V E Sergei Chmutov Partial duality of hypermaps
σ -model for oriented hypermaps v f e σ σ σ V E F Sergei Chmutov Partial duality of hypermaps
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
σ -model. Example. 3 12 1 7 5 8 Sergei Chmutov Partial duality of hypermaps
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
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
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
Duality for graphs G Sergei Chmutov Partial duality of hypermaps
Duality for graphs G Sergei Chmutov Partial duality of hypermaps
Duality for graphs G Sergei Chmutov Partial duality of hypermaps
Duality for graphs G ∗ = G { 1 , 2 , 3 , 4 , 5 , 6 } G Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs G Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs G 1 3 6 4 5 2 Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs G { 1 , 2 , 3 , 4 , 5 } = ??? G 1 3 6 4 5 2 Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) Sergei Chmutov Partial duality of hypermaps
Partial duality for graphs (continuation) R { 1 , 2 , 3 , 4 , 5 } Sergei Chmutov Partial duality of hypermaps
Partial duality for hypermaps Let S be a subset of the vertex-cells of G . Sergei Chmutov Partial duality of hypermaps
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
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
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
Partial duality for hypermaps (continuation) Step 3. Gluing the vertex-cells. Sergei Chmutov Partial duality of hypermaps
Partial duality for hypermaps (continuation) Step 3. Gluing the vertex-cells. Sergei Chmutov Partial duality of hypermaps
Partial duality for hypermaps (continuation) Step 4. Forming the partial dual hypermap G S . Sergei Chmutov Partial duality of hypermaps
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
Partial duality. Properties. (a) The resulting hypermap does not depend on the choice of type at the beginning. Sergei Chmutov Partial duality of hypermaps
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
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
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
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
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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