Matroids, graphs in surfaces, and the Tutte polynomial 2016 International Workshop on Structure in Graphs and Matroids Iain Moffatt and Ben Smith Royal Holloway, University of London Eindhoven, 29 th July 2016
Overview T utte polynomial T opological extensions A matroidal setting Matroid ◮ Introduce matroidal analogues of various notions of polynomials embedded graphs. Graphical analogues ◮ Introduce by applications to the theory of the T utte Unifying T opological T utte polynomials polynomial: 1. Extensions of the T utte polynomial to graphs in surfaces. 2. Incomplete aspects of the theory. 3. � matroid model. 4. T opological graphs ↔ matroid models 13
A review of the T utte polynomial The Tutte polynomial , T ( G ; x , y ) 2 T utte polynomial T opological Polynomial valued graph invariant, T : Graphs → Z [ x , y ] . extensions A matroidal setting ◮ Importance due to applications / combinatorial info. Matroid polynomials (colourings, flows, orientations, codes, Sandpile model, Graphical analogues Potts & Ising models (statistical physics), QFT, Jones & Unifying T opological homflypt polynomials (knot theory), ...) T utte polynomials Definition (deletion-contraction) 1 if G edgeless xT ( G / e ) if e a bridge T ( G ; x , y ) = yT ( G \ e ) if e a loop T ( G \ e ) + T ( G / e ) otherwise 13
A review of the T utte polynomial State sum formulation ( T ( G ) is well-defined) 2 T utte polynomial T opological extensions � ( x − 1 ) r ( G ) − r ( A ) ( y − 1 ) | A |− r ( A ) T ( G ) = A matroidal setting A ⊆ E Matroid polynomials where r ( A ) = # verts . − # cpts . of ( V , A ) = rank of A . Graphical analogues Unifying T opological T utte polynomials ◮ T is defined for matroids (e.g., r = rank function). ◮ T ( C ( G )) = T ( G ) , where C ( G ) is cycle matroid ◮ Matroids often ‘complete’ graph results (e.g. duality) 13
Graphs in surfaces ◮ Plane graph - drawn on a sphere, edges don’t T utte polynomial meet, faces are disks. 3 T opological extensions A matroidal setting Matroid polynomials Graphical analogues ◮ Embedded graph = graph in surface - drawn on Unifying T opological T utte polynomials surface, edges don’t meet. ◮ Cellularly embedded graph - drawn on surface, faces are disks. 13
A T opological T utte polynomial T utte polynomial The Bollobás-Riordan-Krushkal polynomial 4 T opological extensions x r ( G ) − r ( A ) y | A |− r ( A ) a γ ( A ) b γ ∗ ( A c ) � K ( G ; x , y , a , b ) := A matroidal setting Matroid A ⊆ E ( G ) polynomials γ ( A ) := Euler genus of nbhd. of subgraph of G on A Graphical analogues γ ∗ ( A c ) := Euler genus of nbhd. of subgraph of G ∗ on A c Unifying T opological T utte polynomials ◮ T ( G ; x , y ) = K ( G ; x − 1 , y − 1 , 1 , 1 ) ◮ G plane graph = ⇒ T ( G ; x , y ) = K ( G ; x − 1 , y − 1 , a , b ) . 13
A T opological T utte polynomial T utte polynomial 4 T opological extensions ◮ Deletion-contraction definition of the topological A matroidal setting Matroid T utte polynomial: polynomials Graphical analogues Unifying T opological T utte polynomials 13
A T opological T utte polynomial T utte polynomial 4 T opological extensions ◮ Deletion-contraction definition of the topological A matroidal setting T utte polynomial: Matroid polynomials Graphical analogues Unifying T opological T utte polynomials ◮ No (full) recursive definition. 13
A T opological T utte polynomial T utte polynomial ◮ Deletion-contraction definition of the topological 4 T opological T utte polynomial: extensions A matroidal setting Matroid polynomials Graphical analogues Unifying T opological T utte polynomials ◮ No (full) recursive definition. ◮ = ⇒ cell. embedded graphs are not the correct framework for the topological T utte polynomial! ◮ What is the correct framework? 13
Look to matroids T utte polynomial T opological ◮ Why does deletion-contraction fail? extensions 5 A matroidal setting Matroid polynomials wants deletion as Graphical contraction in dual analogues Unifying T opological T utte polynomials ¿ contract ? wants graph contraction wants ribbon graph contraction ◮ Exponents demand incompatible notions of deletion and contraction.... 13
Look to matroids ◮ Why does deletion-contraction fail? T utte polynomial T opological extensions 5 A matroidal setting wants deletion as Matroid Bond matroid, B(G*) contraction in dual polynomials Graphical analogues Unifying T opological ¿ contract ? T utte polynomials wants graph contraction Cycle matroid, C(G) wants ribbon graph contraction Delta-matroid, D(G) ◮ Exponents demand incompatible notions of deletion and contraction.... ◮ ...but these are provided by various matroids. 13
Delta-matroids Symmetric Exchange Axiom (SEA) : ∀ X , Y ∈ F , if ∃ u ∈ X △ Y , T utte polynomial then ∃ v ∈ X △ Y such that X △{ u , v } ∈ F . T opological extensions matroids (via bases) delta-matroids 6 A matroidal setting M = ( E , B ) M = ( E , F ) Matroid polynomials ◮ B � = ∅ , subsets of E ◮ F � = ∅ , subsets of E Graphical analogues ◮ B satisfies SEA ◮ F satisfies SEA Unifying T opological T utte polynomials ◮ X , Y ∈ B = ◮ X , Y ∈ F = ⇒ | X | = | Y | ⇒ | X | = | Y | Cycle matroid (trees) ∆ -matroid (quasi-trees) M ( G ) = ( E , {{ 2 } , { 3 }} ) D ( G ) = ( E , {{ 1 , 2 , 3 } , { 2 } , { 3 }} ) ◮ D min = ( E , { smallest sets } ) a matroid ◮ D max = ( E , { biggest sets } ) a matroid ◮ D ( G ) min = C ( G ) ◮ D ( G ) max = B ( G ∗ ) = ( C ( G ∗ )) ∗ 13
( matroid, delta-matroid, matroid ) ◮ Associate triple to embedded graph: T utte polynomial T opological extensions 7 A matroidal setting Matroid polynomials ◮ Generally, consider triples Graphical analogues Unifying T opological ( M , D , N ) of ( matroid, delta-matroid, matroid ) T utte polynomials ◮ Deletion & contraction: ( M , D , N ) \ e := ( M \ e , D \ e , N \ e ) , ( M , D , N ) / e := ( M / e , D / e , N / e ) ◮ Important observation: different actions of deletion contraction, ( D min ) / e � = ( D / e ) min , ( D \ e ) max � = ( D max \ e ) . (So we have more than the delta-matroid.) 13
Strong maps and matroid perspectives ◮ There is structure we are not seeing. T utte polynomial ◮ Not all (graphic) triples can arise as minors of T opological extensions ( B ( G ∗ ) , D ( G ) , C ( G )) , ◮ e.g., 12 triples ( M , D , N ) on 1 element, only 5 arise. 8 A matroidal setting Matroid = ⇒ missing conditions. ◮ polynomials Graphical Matroid perspectives analogues Unifying T opological T utte polynomials A matroid perspective , is a pair of matroids ( M , N ) over E such that 1. ⇐ ⇒ every circuit of M is union of circuits of N 2. ⇐ ⇒ every flat of N is a flat of M , 3. ⇐ ⇒ r M ( B ) − r M ( A ) ≥ r N ( B ) − r N ( A ) when A ⊆ B ⊆ E 4. ⇐ ⇒ M = H \ A and N = H / A , for some H on E ⊔ A . ◮ Examples of matroid perspectives ◮ ( B ( G ∗ ) , C ( G )) ◮ ( C ( G ) , C ( H )) where H from G by identifying vertices ◮ ( D max , D min ) where D a delta-matroid 13
∆ -perspectives ∆ -perspectives T utte polynomial T opological An ∆ -perspective is a triple ( M , D , N ) such that extensions 9 A matroidal setting 1. M and N are matroids, and D is a Matroid polynomials delta-matroid over the same set, Graphical analogues 2. ( M , D max ) is a matroid perspective Unifying T opological T utte polynomials 3. ( D min , N ) is a matroid perspective ◮ Example: ( B ( G ∗ ) , D ( G ) , C ( G )) is a ∆ -perspective. Theorem If ( M , D , N ) is an ∆ -perspective, then so are ( M , D , N ) \ e and ( M , D , N ) / e . ( M , D , N ) from cell. embed. graph � its minors are. 13
‘T utte polynomial’ of perspectives ◮ There is a canonical way to construct ‘T utte T utte polynomial polynomials’ of objects (via Hopf algebras). T opological extensions Definition: T utte polynomial of ( M , D , N ) A matroidal setting 10 Matroid polynomials � x r ′ ( E ) − r ′ ( A ) y | A |− r ( A ) a ρ ( A ) − r ′ ( A ) b r ( A ) − ρ ( A ) , K ( M , D , N ) := Graphical analogues A ⊆ E Unifying T opological T utte polynomials where ρ = 1 2 ( r max + r min ) . ◮ Theorems: ◮ Contains Bollobás-Riordan-Krushkal polynomial K ( G ; x , y , a , b ) = b γ ( G ) K (( M , D , N ); x , y , a 2 , b − 2 ) ◮ K ( M , D , N ) has a 6 term deletion-contraction relation. ◮ duality formula, convolution formula, universality,... ◮ ∆ -perspectives correct setting for topological T utte polynomials. ◮ Results that should hold for BRK-polynomial but do not, hold for the matroid version of the polynomial. 13
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