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

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duality medial graphs and polynomials of embedded graphs

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

Jun 10, 2023 150 likes •449 views

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 is a (possibly

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SLIDE 1

Duality, medial graphs and polynomials of embedded graphs

Iain Moffatt joint with Jo Ellis-Monaghan BCC, St Andrews, 9th July 2009

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 1 / 12

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SLIDE 2

Medial graphs

Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R2

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

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SLIDE 3

Medial graphs

Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R2

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

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SLIDE 4

Medial graphs

Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R2

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

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SLIDE 5

Medial graphs

Σ is a (possibly non-orientable) surface. G ⊂ Σ is an embedded graph. ⊂ R2 Gm ⊂ Σ is its medial graph.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 2 / 12

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SLIDE 6

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

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SLIDE 7

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

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SLIDE 8

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

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SLIDE 9

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

slide-10

SLIDE 10

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

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SLIDE 11

Tait Graphs

Problem

If F ⊂ Σ is a 4-regular, embedded graph. Find all G ⊂ Σ such that Gm = F. Fbl ⊂ Σ Fwh ⊂ Σ Fbl and Fwh are the Tait graphs of F.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 3 / 12

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SLIDE 12

Properties of Tait graphs

Theorem (Folklore)

For embedded graphs F, G, with F 4-regular Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 4 / 12

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SLIDE 13

A subtlety

Difficulty

Not all 4-reg. emb. graphs are checker-board colourable. Tait graphs don’t always exist! F = Σ = torus F is not a medial graph

Questions

What can we do when F is not checker-board colourable? Which embedded graphs, G, have Gm ∼ = F as graphs? How do these graphs relate to each other?

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 5 / 12

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SLIDE 14

A subtlety

Difficulty

Not all 4-reg. emb. graphs are checker-board colourable. Tait graphs don’t always exist! F = Σ = torus F is not a medial graph

Questions

What can we do when F is not checker-board colourable? Which embedded graphs, G, have Gm ∼ = F as graphs? How do these graphs relate to each other?

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 5 / 12

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SLIDE 15

Embedded graphs

Cellularly embedded graph

* *

Ribbon graph Arrow presentation

2 3 1 1 2 3

* * * *

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 6 / 12

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SLIDE 16

Embedded graphs

Cellularly embedded graph

* *

Ribbon graph Arrow presentation

2 3 1 1 2 3

2 3 1 1 2 3

1 2 3

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 6 / 12

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SLIDE 17

Cycle family graphs

Cycle family graphs generalize Tait graphs. Replace each

v

with one of

v v

r

v v v v

r

v v

r
r

v v

r

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

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SLIDE 18

Generalizing Tait graphs

The questions we’re answering

Which embedded graphs, G, have Gm ∼ = F as graphs? How do these graphs relate to each other?

Theorem (E-M & M)

G, F emb. graphs and F 4-regular. Then Gm ∼ = F ⇐ ⇒ G a cycle family graph of F. Compare with:

The results about medial graphs that we’re extending

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

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SLIDE 19

Generalizing Tait graphs

The questions we’re answering

Which embedded graphs, G, have Gm ∼ = F as graphs? How do these graphs relate to each other?

Theorem (E-M & M)

G, F emb. graphs and F 4-regular. Then Gm ∼ = F ⇐ ⇒ G a cycle family graph of F. Compare with:

The results about medial graphs that we’re extending

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

slide-20

SLIDE 20

Generalizing Tait graphs

The questions we’re answering

Which embedded graphs, G, have Gm ∼ = F as graphs? How do these graphs relate to each other?

Theorem (E-M & M)

G, F emb. graphs and F 4-regular. Then Gm ∼ = F ⇐ ⇒ G a cycle family graph of F. Compare with:

The results about medial graphs that we’re extending

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 8 / 12

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SLIDE 21

Generalizing Duality

We generalize Poincaré duality.

Twisted dual operations

Define operations τ and δ on an arrow presentation by τ

ei ei
=

ei ei

δ

ei ei
=

ei ei

G =

e1 e2

=

e1 e2 e1 e2

, (τ, 1)(G) = Gτ(e1) =

e1 e2 e1 e2

=

e1 e2

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 9 / 12

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SLIDE 22

Generalizing Duality

We generalize Poincaré duality.

Twisted dual operations

Define operations τ and δ on an arrow presentation by τ

ei ei
=

ei ei

δ

ei ei
=

ei ei

G =

e1 e2

=

e1 e2 e1 e2

, (δ, 1)(G) = Gδ(e1) =

e1 e2 e1 e2

=

e1 e2

.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 9 / 12

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SLIDE 23

Twisted Duality

The ribbon group action

The group τ, δ | τ 2, δ2, (τδ)3n acts on embedded graphs with n ordered edges.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

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SLIDE 24

Twisted Duality

The ribbon group action

The group τ, δ | τ 2, δ2, (τδ)3n acts on embedded graphs with n ordered edges.

Example

e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2

(δ, 1) (τ, 1) (1, τ) (1, δ)

= =

(δ, 1) (τ, 1) (1, τ) (1, δ) (δ, 1) (τ, 1) (1, τ) (1, δ)

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

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SLIDE 25

Twisted Duality

Example

e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2

(δ, 1) (τ, 1) (1, τ) (1, δ)

= =

(δ, 1) (τ, 1) (1, τ) (1, δ) (δ, 1) (τ, 1) (1, τ) (1, δ)

Definition (Twisted dual)

The images of G under the group action (with respect to any edge

rder) are its twisted duals.
I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 10 / 12

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SLIDE 26

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

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12

slide-27

SLIDE 27

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

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12

slide-28

SLIDE 28

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 Gm its medial graph, then { twisted duals of G } = { cycle family graphs of Gm } . Compare with:

The results about medial graphs that we’re extending

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 11 / 12

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SLIDE 29

Summary

We have seen

Gm ∼ = F ⇐ ⇒ G a cycle family graph of F All cycle family graphs are twisted duals. { twisted duals of G } = { cycle family graphs of Gm }

Extending classical relations

Gm = F ⇐ ⇒ G = Fbl or G = Fwh; Fbl = (Fwh)∗; {G, G∗} = {(Gm)bl, (Gm)wh}. See paper (arXiv:0906.5557) for: applications to graph polynomials (Penrose, chromatic, transition, topological Tutte polys); applications to knot theory.

I. Moffatt

(South Alabama) Medial graphs and twisted duals BCC 2009 12 / 12