Where do topological Tutte polynomial come from? Iain Moffatt - PowerPoint PPT Presentation

Where do topological Tutte polynomial come from? Iain Moffatt Carolyn Chun, Jo Ellis-Monaghan, Thomas Krajewski, Steve Noble, Ralf Rueckriemen, Ben Smith, Adrian Tanasa Royal Holloway, University of London Dagstuhl, 13 th June 2016

A review of the T utte polynomial The T utte polynomial, T ( G ; x , y ) ∈ Z [ x , y ] 1 Background Minors  1 if G edgeless Constructing the polynomials    if e a bridge xT ( G / e )  Matroids T ( G ) = Remarks yT ( G \ e ) if e a loop    T ( G \ e ) + T ( G / e ) otherwise  � � = x 2 + x + y E.g., T Theorem ◮ T ( G ) is well-defined. ◮ T ( G ) = � ( x − 1 ) r ( G ) − r ( A ) ( y − 1 ) | A |− r ( A ) A ⊆ E ◮ r ( A ) = # verts . − # cpts . of ( V , A ) . ◮ Defined for matroids - take r to to be rank function. ◮ T ( C ( G )) = T ( G ) , where C ( G ) is cycle matroid 16

Embedded graphs = graphs in surfaces ◮ Plane graph - drawn on a sphere , edges don’t 2 Background meet, faces are disks. Minors Constructing the polynomials Matroids Remarks ◮ embedded graph = graph in surface - drawn on surface, edges don’t meet. ◮ cellularly embedded graph - drawn on surface, faces are disks. 16

Three T opological T utte polynomials 3 Background ◮ M. Las Vergnas’ 1978 polynomial, L ( G ; x , y , z ) Minors Constructing the polynomials ( y − 1 ) z 2 · L ( G ; x , y , 1 / ( y − 1 ) z 2 ) Matroids Remarks � ( x − 1 ) r ( G ) − r ( A ) ( y − 1 ) | A |− r ( A ) z γ G ( A ) − γ G ∗ ( E \ A ) := A ⊆ E ◮ B. Bollobás and O. Riordan’s 2002 polynomial � ( x − 1 ) r ( G ) − r ( A ) y | A |− r ( A ) z γ ( A ) R ( G ; x , y , z ) := A ⊆ E ◮ V. Kruskal’s 2011 polynomial 1 1 2 s ⊥ ( A ) � x r ( G ) − r ( A ) y κ ( A ) a 2 s ( A ) b K ( G ; x , y , a , b ) := A ⊆ E ( G ) 16

The plan 4 Background Minors Constructing the polynomials ◮ Explain how all three polynomials arise as the Matroids “T utte polynomial” of embedded graphs. (Big Remarks picture.) ◮ unified / canonical approach ◮ Want: ◮ deletion-contraction relation ◮ terminates in edgeless graph ◮ Problems: ◮ The definition of deletion and contraction ◮ Cases for the relation (analogues of loop, bridge, ordinary) 16

Contraction for graphs in surfaces Background 5 Minors Constructing the polynomials Matroids Remarks 16

Deletion for graphs in surfaces Background 6 Minors Constructing the polynomials Matroids Remarks 16

Deletion-contraction relations Background 7 Minors Constructing the polynomials Matroids ◮ 2 deletion and 2 contractions � 4 domains Remarks 1. Cellularly embedded graphs in surfaces 2. graphs in surfaces 3. Cellularly embedded graphs in pseudo-surfaces 4. graphs in pseudo-surfaces ◮ � four “T utte polynomials” ◮ Need to recognise these four polynomials. 16

A ribbon graph framework Ribbon graph Background 8 Minors A “topological graph” with Constructing the polynomials ◮ discs for vertices, Matroids ◮ ribbons for edges. Remarks (Up to homeos. that preserve vertex-edge structure and cyclic order at vertices.) ◮ Ribbon graphs describe exactly cellularly embedded graphs. ◮ Ribbon graph deletion / contraction → cell. embed. graph deletion / contraction ← 16

Coloured ribbon graph boundary coloured Background vertex coloured 9 Minors Constructing the polynomials Matroids Remarks ◮ ribbon graph ← → cell. embed. in surface ◮ boundary coloured r.g. ← → graph in surface ◮ vertex coloured r.g. ← → cell. embed. in pseudo-surface ◮ vertex and boundary coloured r.g. ← → graph in pseudo-surface 16

Defining a T utte polynomial Background Minors 10 Constructing the  polynomials 1 if G edgeless  Matroids   xT ( G / e ) if e a bridge  Remarks T ( G ) = if e a loop yT ( G \ e )    T ( G \ e ) + T ( G / e ) otherwise  ◮ We have ◮ deletion and contraction ◮ objects closed under the operations ◮ reduce to edgeless ribbon graph ◮ We need ◮ to identify the cases for the recursive definition ◮ i.e., find analogues of loops and bridges 16

The classical case ◮ 2 graphs on 1 edge: , Background Minors ◮ For e c = E \ e , look at pair 11 Constructing the polynomials ( G \ e c , G / e c ) Matroids Remarks ◮ � � � � , ⇐ ⇒ e bridge , ⇐ ⇒ e loop � � � � , ⇐ ⇒ e ordinary , is impossible ◮ Define � 1 if G edgeless U ( G ) = if e is ( i , j ) a i U ( G \ e ) + b j U ( G / e ) ◮ Then U ( G ) is T utte polynomial: U ( G ) = a | E |− r ( G ) b r ( G ) T ( G ; a b b b + 1 , b l a l + 1 ) l b 16

The topological case Background ◮ Apply canonical construction to topological graphs. Minors ◮ 5 vertex and boundary coloured ribbon graphs: 12 Constructing the polynomials Matroids Remarks ◮ Define a T utte polynomial � 1 if G edgeless U ( G ) = a i U ( G \ e ) + b j U ( G / e ) if e is ( i , j ) ◮ � 4 variables (for well-definedness) ◮ � 6 term deletion-contraction definition ◮ � Krushkal’s polynomial K ( G ; x , y , a , b ) . 16

The topological case continued... ◮ The deletion-contraction invariants for the other Background Minors objects are: 13 Constructing the polynomials graph T ( G , x , y ) R ( G , x , y , 1 / √ xy ) Matroids ribbon graph Remarks vertex col. r.g. R ( G , x , y , z ) R ( G ∗ , x , y , z ) boundary col. r.g. vert. & bound. col. r.g. K ( G ; x , y , a , b ) vert. & bound. col. without r.g. L ( G ; x , y , z ) or graph T ( G , x , y ) R ( G , x , y , 1 / √ xy ) cell. emb. surface cell. emb. pseudo-surface R ( G , x , y , z ) R ( G ∗ , x , y , z ) surface pseudo-surface K ( G ; x , y , a , b ) from pseudo-surface L ( G ; x , y , z ) c.f., Krajewski, Moffatt, & T anasa, arXiv:1508.00814. 16

A matroid framework Background Minors Constructing the polynomials 14 Matroids Remarks Moffatt & Smith, ask; Chun, Moffatt, Noble & Rueckriemen, arXiv:1403.0920, arXiv:1602.01306; Ellis-Monaghan & Moffatt, arXiv:1311.3762; Krajewski, Moffatt, & T anasa, arXiv:1508.00814. 16

Summary Background Minors Constructing the polynomials ◮ Canonical approach to the T utte polynomial. 15 Matroids ◮ 4 types of deletion and contraction for embedded Remarks graphs. ◮ � 4 (or 5 or 3) topological T utte polynomials ◮ “full” recursive definition ◮ recovers Bollobás-Riordan, Las Vergnas, and Krushkal polynomials ◮ All polys recovered from Krushkal by forgetting information. ◮ matroid framework. 16

Background Minors Constructing the polynomials 15 Matroids Remarks Thank You! 16

Background Minors Constructing the polynomials 15 Matroids Remarks 16