An introduction to ribbon graphs Iain Moffatt Royal Holloway, - - PowerPoint PPT Presentation

An introduction to ribbon graphs

Iain Moffatt

Royal Holloway, University of London

ALEA-Network Workshop Bordeaux, 16-18 November 2015

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Contents

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51 2

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51 3

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

What is a ribbon graph?

Ribbon graph

A “topological graph” with

◮ discs for vertices, ◮ ribbons for edges.

Graph parameters

v(G) = #(vertices), e(G) = #(edges), k(G) = #(cpts.)

T

• pological parameters

f(G) = #(boundary components), g(G) = genus γ(G) = Euler genus =

• 2g(G)

if orientable g(G) if non-orientable

Euler’s Formula

v(G) − e(G) + f(G) = 2k(G) − γ(G)

51 4

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

What is a ribbon graph?

◮ Cellularly embedded graph – drawn in surface,

faces are discs.

◮ Ribbon graphs describe cellularly embedded

graphs.

t a k e n e i g h b

• u

r h

• d

T a k e s p i n e d e l e t e f a c e s g l u e i n f a c e s

◮ Considered up to homeomorphisms that preserve

vertex-edge structure and cyclic order at vertices. = =

◮ Warning:

◮ Not embedded in R2 or R3. ◮ No concept of a non-loop edge being “twisted”.

51 5

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Arrow presentations

Edges as arrows

e e e e e e

1 2 3

=

3 3 1 2 2

=

1 3 2 2

= = · · · =

1 1

=

1 2 3 2 1 3

=

1 2 3 2 1 3

Arrow presentation

◮ Set of circles ◮ pairs of arrows on them

51

What is a ribbon graph?

6

Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph?

7

Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Deletion and subgraphs

Edge and vertex deletion

e d g e d e l e t i

• n

vertex deletion

G e v G\v G\e

Ribbon subgraphs

◮ ribbon subgraph – delete edges and vertices. ◮ spanning ribbon subgraph – delete edges. ◮ Ribbon graphs are naturally closed under deletion. ◮ But big changes to corresponding cellularly

embedded graph (which are not naturally closed under deletion).

51

What is a ribbon graph?

8

Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Contraction

◮ Care needs to be taken:

◮ Obvious if e = (u, v) non-loop: make e ∪ u ∪ v vertex. ◮ Doesn’t work if e is a loop.

◮ Three routes to a definition of contraction:

◮ Arrow presentations. ◮ Ribbon graphs. ◮ Cellularly embedded graphs.

Edge contraction

T

• contract e = (u, v):

◮ attach disc to each

boundary-cpt. of v ∪ e ∪ u

◮ remove v ∪ e ∪ u

G G/e

◮ Caution: inconsistent with graph contraction!

51

What is a ribbon graph?

9

Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Ribbon graph minors

Ribbon graph minor

Delete vertices, delete edges, contract edges. An anti-chain of graphs but not of ribbon graphs: B =                       = {B3, B5, B7, · · · }

Conjecture (R.-S. theory for ribbon graphs?)

Every minor-closed family of ribbon graphs is characterised by a finite set of excluded minors.

◮ Why not a special case of Robertson-Seymour

Theory?

51

What is a ribbon graph?

10 Subgraphs and

minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Excluded minor characterisations

Proposition

G is orientable ⇐ ⇒ no

• minor

Sg := {G | genus g + 1, orient G =

• [1 vert., 1 ∂-cpt ])}

Theorem

Orientable G is of genus ≤ g ⇐ ⇒ no minor in So

g

51

What is a ribbon graph?

11 Subgraphs and

minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

Excluded minor characterisations

S2k+1 := {G | γ(G) = 2k + 2, G =

• [1 vert., 1 ∂-cpt ])}

S2k := {G | γ(G) = 2k + 1, G =

• [1 vert., 1 ∂-cpt ])
• r γ(G) = 2k + 2, orient G =
• [1 vert., 1 ∂-cpt ])}

Theorem

G is of Euler genus ≤ g ⇐ ⇒ no minor in Sg

51

What is a ribbon graph? Subgraphs and minors

12 Polynomials

Duality Medial graphs Polynomials II Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors

13 Polynomials

Duality Medial graphs Polynomials II Knot theory Matroids

A review of the T utte polynomial

The T utte polynomial, T(G; x, y) ∈ Z[x, y]

T(G) =          1 if G edgeless xT(G/e) if e a bridge yT(G\e) if e a loop T(G\e) + T(G/e)

• therwise

E.g., T

• = x2 + x + y

Theorem

◮ T(G) is well-defined. ◮ T(G) = A⊆E

(x − 1)r(G)−r(A)(y − 1)|A|−r(A)

◮ T(G) encodes lots of information about a graph and

appears in many places. (e.g., chromatic polynomial, flow polynomial, Ising model, Potts model, Jones polynomial, homfly-pt polynomial,....)

51

What is a ribbon graph? Subgraphs and minors

14 Polynomials

Duality Medial graphs Polynomials II Knot theory Matroids

A “T utte polynomial” for ribbon graphs

◮ What do we mean by a T

utte polynomial?

◮ Universal deletion-contraction invariant.

Universality

U(G) :=          1 if G edgeless xU(G/e) if e a bridge yU(G\e) if e a loop σU(G\e) + τU(G/e)

• therwise

Then U(G) = σ|E|−r(G)τ r(G)T(G; x

τ , y σ) ◮ i, j ∈ {bridge, loop} ◮ e is an (i, j)-egde ⇐

⇒

• G/ec is i

G\ec is j

◮ U(G) = aiU(G\e) + bjU(G/e) if e is (i, j)-edge. ◮ U(G) = a|E|−r(G) l

br(G)

b

T(G; ab

bb + 1, bl al + 1)

51

What is a ribbon graph? Subgraphs and minors

15 Polynomials

Duality Medial graphs Polynomials II Knot theory Matroids

A “T utte polynomial” for ribbon graphs

Apply to ribbon graphs:

◮ i, j ∈

• = {b, o, n}

◮ e is an (i, j)-egde ⇐

⇒

• G/ec is i

G\ec is j

◮ α(G) = aiα(G\e) + bjα(G/e) if e is (i, j)-egde. ◮ α well-defined ⇐

⇒ an = (√abao) and bn = (

• bbbo)

“T utte polynomial” of a ribbon graph

˜ R(G, x, y) =

• A⊆E

(x − 1)ρ(G)−ρ(A)(y − 1)|A|−ρ(A), ρ(A) = r(A) + γ(A)/2

◮ α(G) = a|E|−ρ(G) l

bρ(G)

b

˜ R(G; ab

bb + 1, bl al + 1) ◮ 2-variable specialisation of Bollobás-Riordan

polynomial.

51

What is a ribbon graph? Subgraphs and minors Polynomials

16 Duality

Medial graphs Polynomials II Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors Polynomials

17 Duality

Medial graphs Polynomials II Knot theory Matroids

The geometric dual

The dual G∗ of an embedded graph G

◮ One vertex of G∗ in each face of G. ◮ One edge of G∗ when an edge separates faces in G.

G = = G∗

51

What is a ribbon graph? Subgraphs and minors Polynomials

18 Duality

Medial graphs Polynomials II Knot theory Matroids

The geometric dual

The dual G∗ of a ribbon graph G

◮ Fill in punctures of surface G with vertices of G∗, ◮ then delete vertices of G to get G∗.

G =

embed dual expand expand take ribbon graph redraw

= G∗

51

What is a ribbon graph? Subgraphs and minors Polynomials

19 Duality

Medial graphs Polynomials II Knot theory Matroids

The geometric dual

The dual G∗ of an arrow presentation G

e e e e

51

What is a ribbon graph? Subgraphs and minors Polynomials

20 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals

◮ Duality is a local operation! We can define:

Partial duals

GA, the partial dual of G with respect to A ⊆ E(G): dual only edges in A:

• r

← → ← →

51

What is a ribbon graph? Subgraphs and minors Polynomials

21 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

◮ When does G have a partial dual in a given class? ◮ When is G a partial dual of a plane graph?

join (or connected sum)

G = P ∨ Q if

◮ G = P ⊔ Q ◮ P ∩ Q = {v} ◮ E(P), E(Q) meet v

• n disjoint arcs

P

Q

v v

P

Q

v

P ⊔ Q. P ∨ Q

1-sum

G = P ⊕ Q if

◮ G = P ⊔ Q ◮ P ∩ Q = {v}

P

Q

v v

P

Q

v

P ⊔ Q. A P ⊕ Q

51

What is a ribbon graph? Subgraphs and minors Polynomials

22 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

Q

P

vG

1

vG

2

vG

3

P

vP

1

vP

2

vP

3

Q

vQ

1

vQ

2

vQ

3

G = P ⊕3 Q P ⊂ ΣP and Q ⊂ ΣQ

P ∗

Q

vQ

1

vQ

2

vQ

3

Q

P ∗

P∗ ⊂ ΣP and Q ⊂ ΣQ. (P ⊕n Q)E(P) ⊂ Σ

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

23 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

plane-biseparation

A ⊆ E(G) defines a plane-biseparation if the components of G|A and G|Ac

◮ are plane, ◮ intersect in at most one vertex.

P Q

Theorem

GA plane ⇐ ⇒ A defines a plane-biseparation.

51

What is a ribbon graph? Subgraphs and minors Polynomials

24 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

Theorem

Partial dual of plane graph ⇐ ⇒ no minor isomorphic to , ,

51

What is a ribbon graph? Subgraphs and minors Polynomials

25 Duality

Medial graphs Polynomials II Knot theory Matroids

Partial duals of plane graphs

◮ If G and GA are both plane, how are they related?

Theorem

G plane. G = H1 ∨ H2 ∨ · · · ∨ Hp. GA plane ⇐ ⇒ A =

• i∈I

Hi Dualling a summand:

H1 H2 H1 H∗

2

Theorem

G plane. GA plane ⇐ ⇒ obtained by dualling summands of G

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality

26 Medial graphs

Polynomials II Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality

27 Medial graphs

Polynomials II Knot theory Matroids

Medial graphs

The medial graph Gm of a G

◮ One degree 4 vertex on each edge of G ⊂ Σ. ◮ Add edges by following face boundaries.

G = = Gm

Theorem

Gm = Hm ⇐ ⇒ H ∈ {G, G∗}. Concept of graph equivalence ← → Concept of graph duality

◮ Generate new graph dualities by changing concept

• f graph equaivalence.

◮ Let’s determine duality generated by graph

isomorphism.

◮ Proceed by understanding classical case.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality

28 Medial graphs

Polynomials II Knot theory Matroids

T ait graphs, T(F)

T ait graphs

◮ face 2-colour F, ◮ pick a colour & put vertices in faces of that colour. ◮ edges where faces touch

F = = Fbl F = = Fwh

T ait graphs and medial graphs

◮ Gm = Hm ⇐

⇒ H ∈ T(Gm).

◮ H ∈ T(Gm) ⇐

⇒ H = G or H = G∗.

◮ T

• gether give Gm = Hm ⇐

⇒ H ∈ {G, G∗}

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality

29 Medial graphs

Polynomials II Knot theory Matroids

Cycle family graphs, C(F)

◮ Consider T

ait graphs as arrow presentations.

◮ Then drop the restrictions:

Cycle family graphs

Replace each

v

with one of

v v

• r

v v v v

• r

v v

• r
• r

v v

• r

v v

C

• =
• ,

,

• Theorem

Gm ∼ = F ⇐ ⇒ G ∈ C(F)

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What is a ribbon graph? Subgraphs and minors Polynomials Duality

30 Medial graphs

Polynomials II Knot theory Matroids

T wisted duals

Partial Petrial, Gτ(A)

“half-twist” edges is A:

T wisted dual

Twisted dual: a result of a sequence of partial duals and partial Petrials.

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)

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What is a ribbon graph? Subgraphs and minors Polynomials Duality

31 Medial graphs

Polynomials II Knot theory Matroids

T wisted duals

Theorem

C(Gm) = {twisted duals of G}

Theorem

Gm = Hm ⇐ ⇒ H a twisted dual of G Concept of graph equivalence ← → Concept of graph duality equal as embedded graphs ← → Geometric duality equal as abstract graphs ← → T wisted duality equal as rotation systems ← → partial duality

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs

32 Polynomials II

Knot theory Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs

33 Polynomials II

Knot theory Matroids

If only...

OK, I don’t really have time for this :-( But here I would talk about:

◮ The Penrose polynomial. ◮ Various combinatorial interpretations of graph

polynomials.

◮ How they are related via the transition polynomial

and twisted duality.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

34 Knot theory

Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

35 Knot theory

Matroids

The basics of knot theory

◮ knot: is a circle in R3. ◮ link: is disjoint circles in R3. ◮ Considered up to isotopy: “you can push them

around in space”.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

36 Knot theory

Matroids

The basics of knot theory

◮ link diagram: “nice drawing of link on plane” ◮ Reidemeister’s Theorem: L = L′ ⇐

⇒ diagrams related Reidemeister moves

Knot diagrams

project onto plane

Reidemeister moves

= = =

RI RII RIII

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

37 Knot theory

Matroids

The Jones polynomial

◮ Knot invariant: f s.t. f(L) = f(L′) =

⇒ L = L′.

The Kauffman bracket

◮

A =

< > < > < >

+A

• 1

◮ O ∪ L = (−A2 − A−2)L ◮ O = 1

=A +A

• 1

< > < > < >

=A +

< > < >

2

+<

>

A + <

>

• 2

=

< > < >

(A +A )

2

• 2

+ 2 = -A -A

2

• 2

The Jones polynomial

J(L) =

• (−A)−3ω(L)L
• t1/2=A−2

J

• = −t − t2

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

38 Knot theory

Matroids

The ribbon graphs of a link diagram

The ribbon graphs of a link diagram

+/- +/- e e

• r

+/- e e +/-

• +

+ +

• a

b c d e f g

+

• +

+

All-A ribbon graph

A(G):

+/- +/- e e 1 2 3 4 5 6 8 7 1 2 3 4 5 6 7 8 8 1 2 3 4 5 6 7 8

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

39 Knot theory

Matroids

The Jones polynomial as a graph poly- nomial

Recall

◮ T

utte polynomial of a graph: T(G) =

A⊆E

(x − 1)r(G)−r(A)(y − 1)|A|−r(A)

◮ T

utte polynomial of a ribbon graph: ˜ R(G, x, y) =

A⊆E

(x − 1)ρ(G)−ρ(A)(y − 1)|A|−ρ(A).

Theorem

J(D; t) = (−1)w(D)t(3w(D)+|E|−2ρ(A))/4 (−t1/2 − t−1/2)k(A)−γ(A)/2−1˜ R(G, −t, −t−1)

Corollary

For D alternating J(D; t) = (−1)w(D)t(3w(D)+|E|−2r(A))/4 (−t1/2 − t−1/2)k(A)−1˜ T(G, −t, −t−1)

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

40 Knot theory

Matroids

The ribbon graphs of links

◮ How are the ribbon graphs of a link diagram

related?

◮ Which ribbon graphs describe links? ◮ How are diagrams with the same ribbon graphs

related?

Theorem

Ribbon graphs G and G′ represent the same link diagram ⇐ ⇒ they are partial duals.

Theorem

Ribbon graph represents a link diagram ⇐ ⇒ it is partial dual of a plane graph.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

41 Knot theory

Matroids

The ribbon graphs of links

◮ Excluded minors for partial duals of plane graphs

gives:

Theorem

G represents link diagram ⇐ ⇒ no minor isomorphic to , ,

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

42 Knot theory

Matroids

The ribbon graphs of links

◮ Which links have the same ribbon graph? ◮ dualling summands to move between plane partial

duals gives:

Theorem

D and D′ represented by same set of ribbon graphs ⇐ ⇒ related by

a b c a b c

D D′

a b c a b c

D D′

◮ Isotopy class of a link is represented by a unique

set of ribbon graphs.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II

43 Knot theory

Matroids

Relating link digrams

◮ G and GA both plane =

⇒ related by

H1 H2 H1 H∗

2

◮ Embed in S2 and look at link diagram.

v

H1

a b c a b c

H2 H1

c c

v

H∗

2

a b c a b c

D(H1) D(H2)

c c

v

D(H1) D(H∗

2)

.

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

44 Matroids

Plan

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory Matroids

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

45 Matroids

Matroids

◮ Vector spaces matroids

The Symmetric Exchange Axiom (SEA)

◮ ∀X, Y ∈ F, where F a family of sets ◮ u ∈ X△Y =

⇒ ∃v ∈ X△Y s.t. {u, v}△X ∈ F.

matroids (via bases)

M = (E, B)

◮ B = ∅, subsets of E ◮ B satisfies SEA ◮ X, Y ∈ B =

⇒ |X| = |Y|

Graphic matroid

◮ G = (V, E) a graph ◮ B = {edge sets of spanning trees} ◮ cycle matroid of G: M(G) = (E, B)

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

46 Matroids

Ribbon graphs and delta-matorids

◮ Cycle matroids don’t see topology. ◮ Want topological analogue of cycle matroid.

trees

◮ 1 boundary cpt. ◮ genus 0

quasi-trees

◮ 1 boundary cpt.

1 3 2

M(G) = (E, {{2}, {3}})

1 3 2

D(G) = (E, {{1, 2, 3}, {2}, {3}})

D(G)

◮ ribbon graph G = (V, E) ◮ F = {edge sets of spanning quasi-trees} ◮ D(G) = (E, F)

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

47 Matroids

Ribbon graphs and delta-matorids

matroids (via bases)

M = (E, B)

◮ B = ∅, subsets of E ◮ B satisfies SEA ◮ X, Y ∈ B =

⇒ |X| = |Y| Cycle matroid (trees)

1 3 2

M(G) = (E, {{2}, {3}})

delta-matroids

D = (E, F)

◮ F = ∅, subsets of E ◮ F satisfies SEA ◮ X, Y ∈ F =

⇒ |X| = |Y| ∆-matroid (quasi-trees)

1 3 2

D(G) = (E, {{1, 2, 3}, {2}, {3}})

Theorem

D(G) is a delta-matroid. Another example: non-singular principal submatrices.

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What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

48 Matroids

Basic properties

• max. min. matroids of D

◮ Fmin = {feasible sets of min size} ◮ Fmax = {feasible sets of max size} ◮ Dmin = (E, Fmin) a matroid. ◮ Dmax = (E, Fmax) a matroid. ◮ D(G)min = M(G) ◮ D(G)max = M(G∗)∗ ◮ γ(G) = r(D(G)max) − r(D(G)min) ◮ G plane ⇐

⇒ D(G) = M(G)

◮ G orientable ⇐

⇒ all feasible sets have same parity

51

What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

49 Matroids

Polynomials

T utte polynomial

◮ M = (E, B) has rank function r ◮ T(M; x, y) = A⊆E

(x − 1)r(M)−r(A)(y − 1)|A|−r(A)

◮ T(G) = T(M(G))

topological T utte polynomial

◮ Dmin rank function rmin; Dmax rank function rmax ◮ Average rank functions: ρ := 1 2(rmax + rmin). ◮ ˜

R(D; x, y) :=

A⊆E

(x − 1)ρ(D)−ρ(A)(y − 1)|A|−ρ(A)

◮ ˜

R(G) = ˜ R(D(G)) graph polynomial ← → matroid polynomial topological graph polynomial ← → delta-matroid polynomial

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What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

50 Matroids

T wists of matroids

◮ Arguably the most fundamental operation in

delta-matroid theory.

T wists

D ∗ A = (E, {F △ A | F ∈ F}) Example: M = ({1, 2, 3}, {{1}, {2}}) M∗ = M ∗ E = ({1, 2, 3}, {{2, 3}, {1, 3}}) M ∗ {2, 3} = ({1, 2, 3}, {{1, 2, 3}, {3}})

Facts

◮ D∗ = D ∗ E ◮ G plane, (M(G))∗ = M(G∗) ◮ All G, (D(G))∗ = D(G∗) ◮ D(G) ∗ A = D(GA)

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What is a ribbon graph? Subgraphs and minors Polynomials Duality Medial graphs Polynomials II Knot theory

51 Matroids

T wists of matroids

◮ When is D a twist of a matroid? ◮ Translate into ribbon graphs.

Ribbon graph version

Partial dual of plane graph ⇐ ⇒ no minor isomorphic to , ,

Theorem

Partial dual of matroid ⇐ ⇒ no minor isomorphic to D     , D       , D    

◮ Ribbon graphs guide delta-matroid theory.