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

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

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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

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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

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Maps (Graphs on surfaces)

Sergei Chmutov Partial duality of hypermaps

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Maps (Graphs on surfaces)

Sergei Chmutov Partial duality of hypermaps

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Maps (Graphs on surfaces)

Sergei Chmutov Partial duality of hypermaps

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Maps (Graphs on surfaces)

Sergei Chmutov Partial duality of hypermaps

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Hypermaps

Sergei Chmutov Partial duality of hypermaps

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Hypermaps

Sergei Chmutov Partial duality of hypermaps

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Hypermaps

Sergei Chmutov Partial duality of hypermaps

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Hypermaps

Sergei Chmutov Partial duality of hypermaps

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Hypermaps

Sergei Chmutov Partial duality of hypermaps

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τ-model for hypermaps

a face a flag a (hyper) edge a vertex

Sergei Chmutov Partial duality of hypermaps

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τ-model for hypermaps

a face a flag a (hyper) edge a vertex v’ (v,e,f) (v’,e,f) e f v τ

Sergei Chmutov Partial duality of hypermaps

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τ-model for hypermaps

a face a flag a (hyper) edge a vertex v’ (v,e,f) (v’,e,f) e f v τ

1

τ (v,e’,f) (v,e,f) e v e’ f

Sergei Chmutov Partial duality of hypermaps

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τ-model for hypermaps

a face a flag a (hyper) edge a vertex v’ (v,e,f) (v’,e,f) e f v τ

1

τ (v,e’,f) (v,e,f) e v e’ f

(v,e,f’) (v,e,f)

2

τ v e f’ f

Sergei Chmutov Partial duality of hypermaps

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τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

Sergei Chmutov Partial duality of hypermaps

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τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7)

Sergei Chmutov Partial duality of hypermaps

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τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7) τ1 = (1, 2)(3, 4)(5, 6)(7, 9)(8, 10)(11, 12)

Sergei Chmutov Partial duality of hypermaps

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τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7) τ1 = (1, 2)(3, 4)(5, 6)(7, 9)(8, 10)(11, 12) τ2 = (1, 6)(2, 3)(4, 5)(7, 11)(8, 9)(10, 12)

Sergei Chmutov Partial duality of hypermaps

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σ-model for oriented hypermaps

v

V

σ

Sergei Chmutov Partial duality of hypermaps

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σ-model for oriented hypermaps

v

V

σ

E

e σ

Sergei Chmutov Partial duality of hypermaps

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σ-model for oriented hypermaps

v

V

σ

E

e σ

F

f σ

Sergei Chmutov Partial duality of hypermaps

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σ-model for oriented hypermaps

v

V

σ

E

e σ

F

f σ

σFσEσV = 1 :

F

σV σE σ

Sergei Chmutov Partial duality of hypermaps

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σ-model. Example.

8 1 3 5 7 12

Sergei Chmutov Partial duality of hypermaps

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σ-model. Example.

8 1 3 5 7 12

σV = (1, 3, 5)(7, 8, 12) = τ2τ1|{1,3,5,7,8,12}

Sergei Chmutov Partial duality of hypermaps

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σ-model. Example.

8 1 3 5 7 12

σV = (1, 3, 5)(7, 8, 12) = τ2τ1|{1,3,5,7,8,12} σE = (1, 7)(3, 12)(5, 8) = τ0τ2|{1,3,5,7,8,12}

Sergei Chmutov Partial duality of hypermaps

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σ-model. Example.

8 1 3 5 7 12

σV = (1, 3, 5)(7, 8, 12) = τ2τ1|{1,3,5,7,8,12} σE = (1, 7)(3, 12)(5, 8) = τ0τ2|{1,3,5,7,8,12} σF = (1, 12)(3, 8)(5, 7) = τ1τ0|{1,3,5,7,8,12}

Sergei Chmutov Partial duality of hypermaps

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Duality for graphs G

Sergei Chmutov Partial duality of hypermaps

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Duality for graphs G

Sergei Chmutov Partial duality of hypermaps

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Duality for graphs G

Sergei Chmutov Partial duality of hypermaps

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Duality for graphs G

G∗ = G{1,2,3,4,5,6}

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs G

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs G

1 2 3 4 5 6

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs G

1 2 3 4 5 6 G{1,2,3,4,5} = ???

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

Sergei Chmutov Partial duality of hypermaps

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Partial duality for graphs (continuation)

R{1,2,3,4,5}

Sergei Chmutov Partial duality of hypermaps

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Partial duality for hypermaps

Let S be a subset of the vertex-cells of G.

Sergei Chmutov Partial duality of hypermaps

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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

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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

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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 GS.

Sergei Chmutov Partial duality of hypermaps

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Partial duality for hypermaps (continuation)

Step 3. Gluing the vertex-cells.

Sergei Chmutov Partial duality of hypermaps

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Partial duality for hypermaps (continuation)

Step 3. Gluing the vertex-cells.

Sergei Chmutov Partial duality of hypermaps

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Partial duality for hypermaps (continuation)

Step 4. Forming the partial dual hypermap GS.

Sergei Chmutov Partial duality of hypermaps

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Partial duality for hypermaps (continuation)

Step 4. Forming the partial dual hypermap GS.

3 7 8 8 1 12 12 5 5 1 1 5 3 7 11 10 8 2 4 6 12 9 Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning.

Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning. (b)

  • GSS = G.

Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning. (b)

  • GSS = G.

(c) There is a bijection between the cells of type S in G and the cells of the same type in GS. This bijection preserves the valency of cells. The number of cell of other types may change.

Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning. (b)

  • GSS = G.

(c) There is a bijection between the cells of type S in G and the cells of the same type in GS. 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 GS∪{s} =

  • GS{s}.

Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning. (b)

  • GSS = G.

(c) There is a bijection between the cells of type S in G and the cells of the same type in GS. 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 GS∪{s} =

  • GS{s}.

(e)

  • GSS′

= G∆(S,S′), where ∆(S, S′) := (S ∪ S′) \ (S ∩ S′) is the symmetric difference of sets.

Sergei Chmutov Partial duality of hypermaps

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Partial duality. Properties.

(a) The resulting hypermap does not depend on the choice of type at the beginning. (b)

  • GSS = G.

(c) There is a bijection between the cells of type S in G and the cells of the same type in GS. 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 GS∪{s} =

  • GS{s}.

(e)

  • GSS′

= G∆(S,S′), where ∆(S, S′) := (S ∪ S′) \ (S ∩ S′) is the symmetric difference of sets. (f) The partial duality preserves orientability of hypermaps.

Sergei Chmutov Partial duality of hypermaps

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Partial duality in τ-model.

  • Theorem. Consider the τ-model for a hypermap G given by the

permutations τ0(G) : (v, e, f) → (v′, e, f), τ1(G : (v, e, f) → (v, e′, f), τ2(G) : (v, e, f) → (v, e, f ′) of its local flags. Let V ′ be a subset of its vertices, τ V ′

1

be the product

  • f all transpositions in τ1 for v ∈ V ′, and τ V ′

2

be the product of all transpositions in τ2 for v ∈ V ′. Then its partial dual GV ′ is given by the permutations τ0(GV ′) = τ0, τ1(GV ′) = τ1τ V ′

1 τ V ′ 2 ,

τ2(GV ′) = τ1τ V ′

1 τ V ′ 2

. In other words the permutations τ1 and τ2 swap their transpositions of local flags around the vertices in V ′. The similar statement hold for partial duality relative to the subset of hyperedges E′ and for a subset of faces F ′.

Sergei Chmutov Partial duality of hypermaps

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Partial duality in τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7) τ1 = (1,2)(3,4)(5,6) (7, 9)(8, 10)(11, 12) τ2 = (1,6)(2,3)(4,5) (7, 11)(8, 9)(10, 12)

Sergei Chmutov Partial duality of hypermaps

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Partial duality in τ-model. Example.

1 2 3 6 4 7 5 11 12 8 10 9

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7) τ1 = (1,2)(3,4)(5,6) (7, 9)(8, 10)(11, 12) τ2 = (1,6)(2,3)(4,5) (7, 11)(8, 9)(10, 12)

8 2 4 6 12 9 1 5 3 7 11 10

τ0 = (1, 11)(2, 12)(3, 10)(4, 8)(5, 9)(6, 7) τ1 = (1,6)(2,3)(4,5) (7, 9)(8, 10)(11, 12) τ2 = (1,2)(3,4)(5,6) (7, 11)(8, 9)(10, 12)

Sergei Chmutov Partial duality of hypermaps

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Partial duality in σ-model.

  • Theorem. Let S be a subsets S := V ′ of vertices (resp. subset
  • f hyperedges S := E′ and subset of faces S := F ′) of a

hypermap G. Then its partial dual is given by the permutations GV ′ = (σV ′σ−1

V ′ , σEσV ′, σV ′σF)

GE′ = (σE′σV, σE′σ−1

E′ , σFσE′)

GF ′ = (σVσF ′, σF ′σE, σF ′σ−1

F ′ ) ,

where σV ′, σE′, σF ′ denote the permutations consisting of cycles corresponding to the elements of V ′, E′, F ′ respectively, and overline means the complementary set of cycles.

Sergei Chmutov Partial duality of hypermaps

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Partial duality in σ-model. Example.

8 1 3 5 7 12

σV = (1, 3, 5)(7, 8, 12) σE = (1, 7)(3, 12)(5, 8) σF = (1, 12)(3, 8)(5, 7)

Sergei Chmutov Partial duality of hypermaps

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Partial duality in σ-model. Example.

8 1 3 5 7 12

σV = (1, 3, 5)(7, 8, 12) σE = (1, 7)(3, 12)(5, 8) σF = (1, 12)(3, 8)(5, 7)

1 5 3 7 8 12

σV(G{v}) = σV ′σ−1

V ′ = (1, 5, 3)(7, 8, 12)

σE(G{v}) = σEσV ′ = (1, 12, 3, 8, 5, 7) σF(G{v}) = σV ′σF = (1, 12, 3, 8, 5, 7)

Sergei Chmutov Partial duality of hypermaps