An introduction to ribbon graphs Iain Moffatt Royal Holloway, University of London ALEA-Network Workshop Bordeaux, 16-18 November 2015
Contents What is a ribbon graph? What is a ribbon graph? Subgraphs and minors Polynomials Subgraphs and minors Duality Medial graphs Polynomials Polynomials II Knot theory Duality Matroids Medial graphs Polynomials II Knot theory Matroids 51
Plan 2 What is a ribbon graph? What is a ribbon graph? Subgraphs and minors Polynomials Subgraphs and minors Duality Medial graphs Polynomials Polynomials II Knot theory Duality Matroids Medial graphs Polynomials II Knot theory Matroids 51
What is a ribbon graph? Ribbon graph 3 What is a ribbon graph? A “topological graph” Subgraphs and minors with Polynomials ◮ discs for vertices, Duality Medial graphs ◮ ribbons for edges. Polynomials II Knot theory Graph parameters Matroids v ( G ) = # (vertices), e ( G ) = # (edges), k ( G ) = # (cpts.) T opological parameters f ( G ) = # (boundary components), g ( G ) = genus � 2 g ( G ) if orientable γ ( G ) = Euler genus = g ( G ) if non-orientable Euler’s Formula v ( G ) − e ( G ) + f ( G ) = 2 k ( G ) − γ ( G ) 51
What is a ribbon graph? 4 What is a ribbon graph? ◮ Cellularly embedded graph – drawn in surface, Subgraphs and minors faces are discs. Polynomials ◮ Ribbon graphs describe cellularly embedded Duality graphs. Medial graphs n e i g h b o u e l e t e f a c e s k e r h d a o t o d Polynomials II Knot theory n e g l s T a k e s p i u e i n f a c e Matroids ◮ Considered up to homeomorphisms that preserve vertex-edge structure and cyclic order at vertices. = = ◮ Warning: ◮ Not embedded in R 2 or R 3 . ◮ No concept of a non-loop edge being “twisted”. 51
Arrow presentations 5 What is a ribbon graph? Edges as arrows Subgraphs and minors Polynomials e e e e e e Duality Medial graphs 1 1 1 Polynomials II 2 2 2 2 2 Knot theory = = = Matroids 3 3 3 3 2 2 2 2 = · · · = = = 1 1 1 1 1 1 3 3 3 3 Arrow presentation ◮ Set of circles ◮ pairs of arrows on them 51
Plan What is a ribbon graph? What is a ribbon graph? 6 Subgraphs and minors Polynomials Subgraphs and minors Duality Medial graphs Polynomials Polynomials II Knot theory Duality Matroids Medial graphs Polynomials II Knot theory Matroids 51
Deletion and subgraphs What is a ribbon Edge and vertex deletion graph? 7 Subgraphs and vertex deletion minors e d g e d e e Polynomials v l e t i o n Duality G Medial graphs Polynomials II G\e Knot theory G\v Matroids 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
Contraction ◮ Care needs to be taken: What is a ribbon ◮ Obvious if e = ( u , v ) non-loop: make e ∪ u ∪ v vertex. graph? ◮ Doesn’t work if e is a loop. 8 Subgraphs and minors ◮ Three routes to a definition of contraction: Polynomials ◮ Arrow presentations. Duality ◮ Ribbon graphs. Medial graphs ◮ Cellularly embedded graphs. Polynomials II Knot theory Edge contraction Matroids G / e G o contract e = ( u , v ) : T ◮ attach disc to each boundary-cpt. of v ∪ e ∪ u ◮ remove v ∪ e ∪ u ◮ Caution: inconsistent with graph contraction! 51
Ribbon graph minors What is a ribbon Ribbon graph minor graph? 9 Subgraphs and Delete vertices, delete edges, contract edges. minors Polynomials An anti-chain of graphs but not of ribbon graphs: Duality Medial graphs Polynomials II Knot theory B = = { B 3 , B 5 , B 7 , · · · } Matroids 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
Excluded minor characterisations What is a ribbon graph? 10 Subgraphs and minors Proposition Polynomials Duality Medial graphs G is orientable ⇐ ⇒ no -minor Polynomials II Knot theory Matroids � S g := { G | genus g + 1 , orient G = [ 1 vert., 1 ∂ -cpt ]) } Theorem Orientable G is of genus ≤ g ⇐ ⇒ no minor in S o g 51
Excluded minor characterisations What is a ribbon graph? 11 Subgraphs and minors Polynomials � S 2 k + 1 := { G | γ ( G ) = 2 k + 2 , G = [ 1 vert., 1 ∂ -cpt ]) } Duality Medial graphs Polynomials II Knot theory � S 2 k := { G | γ ( G ) = 2 k + 1 , G = [ 1 vert., 1 ∂ -cpt ]) Matroids � or γ ( G ) = 2 k + 2 , orient G = [ 1 vert., 1 ∂ -cpt ]) } Theorem G is of Euler genus ≤ g ⇐ ⇒ no minor in S g 51
Plan What is a ribbon graph? What is a ribbon graph? Subgraphs and minors 12 Polynomials Subgraphs and minors Duality Medial graphs Polynomials Polynomials II Knot theory Duality Matroids Medial graphs Polynomials II Knot theory Matroids 51
A review of the T utte polynomial utte polynomial, T ( G ; x , y ) ∈ Z [ x , y ] What is a ribbon The T graph? Subgraphs and 1 if G edgeless minors xT ( G / e ) 13 Polynomials if e a bridge T ( G ) = Duality yT ( G \ e ) if e a loop Medial graphs T ( G \ e ) + T ( G / e ) otherwise Polynomials II Knot theory � � = x 2 + x + y E.g., T Matroids Theorem ◮ T ( G ) is well-defined. ( x − 1 ) r ( G ) − r ( A ) ( y − 1 ) | A |− r ( A ) ◮ T ( G ) = � A ⊆ E ◮ 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
A “T utte polynomial” for ribbon graphs ◮ What do we mean by a T utte polynomial? What is a ribbon graph? ◮ Universal deletion-contraction invariant. Subgraphs and minors Universality 14 Polynomials Duality 1 if G edgeless Medial graphs xU ( G / e ) if e a bridge Polynomials II U ( G ) := yU ( G \ e ) Knot theory if e a loop Matroids σ U ( G \ e ) + τ U ( G / e ) otherwise Then U ( G ) = σ | E |− r ( G ) τ r ( G ) T ( G ; x τ , y σ ) ◮ i , j ∈ { bridge , loop } � G / e c is i ◮ e is an ( i , j ) -egde ⇐ ⇒ G \ e c is j ◮ U ( G ) = a i U ( G \ e ) + b j U ( G / e ) if e is ( i , j ) -edge. ◮ U ( G ) = a | E |− r ( G ) b r ( G ) T ( G ; a b b b + 1 , b l a l + 1 ) l b 51
A “T utte polynomial” for ribbon graphs Apply to ribbon graphs: What is a ribbon � � graph? ◮ i , j ∈ = { b , o , n } Subgraphs and minors � G / e c is i 15 Polynomials ◮ e is an ( i , j ) -egde ⇐ ⇒ G \ e c is j Duality Medial graphs ◮ α ( G ) = a i α ( G \ e ) + b j α ( G / e ) if e is ( i , j ) -egde. Polynomials II ⇒ a n = ( √ a b a o ) and b n = ( � ◮ α well-defined ⇐ b b b o ) Knot theory Matroids “T utte polynomial” of a ribbon graph ˜ � ( x − 1 ) ρ ( G ) − ρ ( A ) ( y − 1 ) | A |− ρ ( A ) , R ( G , x , y ) = A ⊆ E ρ ( A ) = r ( A ) + γ ( A ) / 2 ◮ α ( G ) = a | E |− ρ ( G ) b ρ ( G ) ˜ R ( G ; a b b b + 1 , b l a l + 1 ) l b ◮ 2-variable specialisation of Bollobás-Riordan polynomial. 51
Plan What is a ribbon graph? What is a ribbon graph? Subgraphs and minors Polynomials Subgraphs and minors 16 Duality Medial graphs Polynomials Polynomials II Knot theory Duality Matroids Medial graphs Polynomials II Knot theory Matroids 51
The geometric dual What is a ribbon graph? Subgraphs and minors The dual G ∗ of an embedded graph G Polynomials 17 Duality ◮ One vertex of G ∗ in each face of G . Medial graphs ◮ One edge of G ∗ when an edge separates faces in G . Polynomials II Knot theory Matroids G = = G ∗ 51
The geometric dual What is a ribbon graph? Subgraphs and minors The dual G ∗ of a ribbon graph G Polynomials ◮ Fill in punctures of surface G with vertices of G ∗ , 18 Duality Medial graphs ◮ then delete vertices of G to get G ∗ . Polynomials II Knot theory embed dual expand Matroids G = take expand ribbon redraw = G ∗ graph 51
The geometric dual The dual G ∗ of an arrow presentation G What is a ribbon graph? Subgraphs and e minors e Polynomials e e 19 Duality Medial graphs Polynomials II Knot theory Matroids 51
Partial duals ◮ Duality is a local operation! We can define: What is a ribbon graph? Partial duals Subgraphs and minors G A , the partial dual of G with respect to A ⊆ E ( G ) : Polynomials 20 Duality dual only edges in A : Medial graphs Polynomials II Knot theory Matroids or ← → ← → 51
Partial duals of plane graphs ◮ When does G have a partial dual in a given class? What is a ribbon graph? ◮ When is G a partial dual of a plane graph? Subgraphs and minors Polynomials join (or connected sum) 21 Duality Medial graphs G = P ∨ Q if Polynomials II ◮ G = P ⊔ Q Knot theory P v v Q P v Q Matroids ◮ P ∩ Q = { v } P ⊔ Q . P ∨ Q ◮ E ( P ) , E ( Q ) meet v on disjoint arcs 1-sum G = P ⊕ Q if v Q P v Q P v ◮ G = P ⊔ Q ◮ P ∩ Q = { v } P ⊔ Q . A P ⊕ Q 51
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