Duality, medial graphs and polynomials of embedded graphs Iain - PowerPoint PPT Presentation

Duality, medial graphs and polynomials of embedded graphs Iain Moffatt joint with Jo Ellis-Monaghan BCC, St Andrews, 9 th July 2009 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 1 / 12

Medial graphs Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

Medial graphs Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

Medial graphs Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

Medial graphs Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R 2 G m ⊂ Σ is its medial graph. I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Tait Graphs Problem If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that G m = F . F wh ⊂ Σ F bl ⊂ Σ F bl and F wh are the Tait graphs of F . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

Properties of Tait graphs Theorem (Folklore) For embedded graphs F, G, with F 4-regular G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 4 / 12

A subtlety Difficulty Not all 4-reg. emb. graphs are checker-board colourable. Tait graphs don’t always exist! Σ = torus F = F is not a medial graph Questions What can we do when F is not checker-board colourable? Which embedded graphs, G , have G m ∼ = F as graphs? How do these graphs relate to each other? I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 5 / 12

A subtlety Difficulty Not all 4-reg. emb. graphs are checker-board colourable. Tait graphs don’t always exist! Σ = torus F = F is not a medial graph Questions What can we do when F is not checker-board colourable? Which embedded graphs, G , have G m ∼ = F as graphs? How do these graphs relate to each other? I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 5 / 12

Embedded graphs Cellularly Ribbon graph Arrow presentation embedded graph 1 1 3 3 * * 2 2 * * * * I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 6 / 12

Embedded graphs Cellularly Ribbon graph Arrow presentation embedded graph 1 1 3 3 * * 2 2 1 1 1 3 3 3 2 2 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 6 / 12

Cycle family graphs Cycle family graphs generalize Tait graphs. v Replace each with one of v v v v v v or or or or or v v v v v v Gives arrow presentation of a cycle family graph of F . = = I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 7 / 12

Generalizing Tait graphs The questions we’re answering Which embedded graphs, G , have G m ∼ = F as graphs? How do these graphs relate to each other? Theorem (E-M & M) G, F emb. graphs and F 4 -regular. Then G m ∼ = F ⇐ ⇒ G a cycle family graph of F . Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

Generalizing Tait graphs The questions we’re answering Which embedded graphs, G , have G m ∼ = F as graphs? How do these graphs relate to each other? Theorem (E-M & M) G, F emb. graphs and F 4 -regular. Then G m ∼ = F ⇐ ⇒ G a cycle family graph of F . Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

Generalizing Tait graphs The questions we’re answering Which embedded graphs, G , have G m ∼ = F as graphs? How do these graphs relate to each other? Theorem (E-M & M) G, F emb. graphs and F 4 -regular. Then G m ∼ = F ⇐ ⇒ G a cycle family graph of F . Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

Generalizing Duality We generalize Poincaré duality. Twisted dual operations Define operations τ and δ on an arrow presentation by � � � � e i = = τ δ e i e i e i e i e i e i e i e 1 e 1 e 1 G = = , e 2 e 2 e 2 e 1 e 1 e 1 ( τ, 1 )( G ) = G τ ( e 1 ) = = e 2 e 2 e 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 9 / 12

Generalizing Duality We generalize Poincaré duality. Twisted dual operations Define operations τ and δ on an arrow presentation by � � � � e i τ = δ = e i e i e i e i e i e i e i e 1 e 1 e 1 G = = , e 2 e 2 e 2 e 2 e 1 ( δ, 1 )( G ) = G δ ( e 1 ) = = . e 1 e 2 e 2 e 1 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 9 / 12

Twisted Duality The ribbon group action The group � τ, δ | τ 2 , δ 2 , ( τδ ) 3 � n acts on embedded graphs with n ordered edges. I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

Twisted Duality The ribbon group action The group � τ, δ | τ 2 , δ 2 , ( τδ ) 3 � n acts on embedded graphs with n ordered edges. Example (1 , τ ) e 2 e 2 e 2 ( δ , 1) ( τ , 1) (1 , δ ) ( τ , 1) (1 , τ ) e 1 e 1 e 1 ( δ , 1) (1 , δ ) e 1 e 1 e 1 e 1 e 2 ( δ , 1) ( τ , 1) = = (1 , δ ) (1 , τ ) e 2 e 1 e 2 e 2 e 2 I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

Twisted Duality Example (1 , τ ) e 2 e 2 e 2 ( δ , 1) ( τ , 1) (1 , δ ) ( τ , 1) (1 , τ ) e 1 e 1 e 1 ( δ , 1) (1 , δ ) e 1 e 1 e 1 e 1 e 2 ( δ , 1) ( τ , 1) = = (1 , δ ) (1 , τ ) e 2 e 1 e 2 e 2 e 2 Definition (Twisted dual) The images of G under the group action (with respect to any edge order) are its twisted duals. I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

Cycle family graphs and twisted duals The questions we’re answering Which embedded graphs have a medial graph isomorphic to F ? How do these graphs relate to each other? Theorem (E-M & M) If F is a 4 -regular embedded graph, then all cycle family graphs are twisted duals. Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12

Cycle family graphs and twisted duals The questions we’re answering Which embedded graphs have a medial graph isomorphic to F ? How do these graphs relate to each other? Theorem (E-M & M) If F is a 4 -regular embedded graph, then all cycle family graphs are twisted duals. Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12

Cycle family graphs and twisted duals Theorem (E-M & M) If F is a 4 -regular embedded graph, then all cycle family graphs are twisted duals. Theorem (E-M & M) If G is embedded and G m its medial graph, then { twisted duals of G } = { cycle family graphs of G m } . Compare with: The results about medial graphs that we’re extending G m = F ⇐ ⇒ G = F bl or G = F wh ; F bl = ( F wh ) ∗ ; { G , G ∗ } = { ( G m ) bl , ( G m ) wh } . I. Moffatt (South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12