B -symmetric chromatic function of signed Sergei graphs Chmutov - PowerPoint PPT Presentation

B -symmetric chromatic function B -symmetric chromatic function of signed Sergei graphs Chmutov Stanley’s chromatic symmetric function. Sergei Chmutov Vassiliev knot invariants. Ohio State University, Mansfield Signed graphs Weighted signed with chromatic function. James Enouen Bialgebra of Eric Fawcett doubly weighted Rushil Raghavan signed graphs. Ishaan Shah Open problems. Thursday, May 7, 2020 Combinatorics of Vassiliev knot invariants seminar National Research University, Higher School of Economics, Moscow (Russia)

B -symmetric chromatic Overview function Sergei Chmutov Stanley’s 1 Stanley’s chromatic symmetric function. chromatic symmetric function. Vassiliev knot Vassiliev knot invariants. 2 invariants. Signed graphs Weighted Signed graphs 3 signed chromatic function. 4 Weighted signed chromatic function. Bialgebra of doubly weighted signed graphs. Bialgebra of doubly weighted signed graphs. 5 Open problems. Open problems. 6

B -symmetric chromatic function Stanley’s chromatic symmetric function. Sergei Chmutov R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph , Advances in Math. 111 (1) Stanley’s chromatic 166–194 (1995). symmetric function. � � Vassiliev knot X G ( x 1 , x 2 , . . . ) := x κ ( v ) invariants. Signed graphs κ : V ( G ) → N v ∈ V ( G ) proper Weighted signed chromatic ∞ � function. x m Power function basis . p m := i . Bialgebra of doubly i = 1 weighted signed graphs. � Example. X = x 1 x 1 + x 1 x 2 + x 1 x 3 + . . . Open x 2 x 1 + � x 2 x 2 + x 2 x 3 + . . . problems. x 3 x 1 + x 3 x 2 + � x 3 x 3 + . . . . . ... . . . . p 2 = 1 − p 2 .

B -symmetric chromatic function Chromatic symmetric function in power basis. Sergei Chmutov � � � Stanley’s X G ( x 1 , x 2 , ... )= x κ ( v ) ( 1 − δ κ ( v 1 ) , κ ( v 2 ) ) chromatic symmetric function. κ : V ( G ) → N v ∈ V ( G ) e =( v 1 , v 2 ) ∈ E ( G ) all � � � ( − 1 ) | S | � Vassiliev knot invariants. = x κ ( v ) δ κ ( v 1 ) , κ ( v 2 ) Signed graphs κ : V ( G ) → N v ∈ V ( G ) S ⊆ E G e ∈ S all Weighted  signed 1  all vertices of a connected component of the span- � chromatic function. ning subgraph with S edges are colored by κ into δ κ ( v 1 ) , κ ( v 2 ) =  the same color Bialgebra of 0 e ∈ S doubly otherwise weighted signed � graphs. ( − 1 ) | S | p λ ( S ) , X G = Open problems. S ⊆ E G where λ ( S ) ⊢ | V ( G ) | is a partition of the number of verticies according to the connected components of the spanning subgraph S , and for λ ( S ) = ( λ 1 , . . . , λ k ) , p λ ( S ) := p λ 1 p λ 2 . . . p λ k .

B -symmetric chromatic function Chromatic symmetric function. Examples. Sergei Chmutov � Stanley’s chromatic ( − 1 ) | S | p λ 1 p λ 2 . . . p λ k X G = symmetric function. S ⊆ E G Vassiliev knot invariants. = p 2 Examples. X 1 − p 2 , Signed graphs Weighted = p 3 = p 3 signed X 1 − 2 p 1 p 2 + p 3 , X 1 − 3 p 1 p 2 + 2 p 3 . chromatic function. p 4 1 − 3 p 2 1 p 2 + p 2 Bialgebra of X = 2 + 2 p 1 p 3 − p 4 , doubly weighted signed p 4 1 − 3 p 2 X = 1 p 2 + 3 p 1 p 3 − p 4 . graphs. Open problems. Two graphs with the same chromatic symmetric function: = X X

B -symmetric chromatic function Chromatic symmetric function. Conjectures. Sergei Chmutov Stanley’s chromatic Tree conjecture. symmetric function. X G distingushes trees. Vassiliev knot invariants. Signed graphs A ( 3 + 1 ) poset is the disjoint union of a 3-element chain and 1-element chain. Weighted A poset P is ( 3 + 1 ) - free if it contains no induced ( 3 + 1 ) posets. signed chromatic Incomparability graph inc ( P ) of P : vertices are elements of P ; ( uv ) is an edge if function. neither u � v nor v � u . Bialgebra of doubly weighted signed e -positivity conjecture. graphs. The expansion of X inc ( P ) in terms of elementary symmetric Open problems. functions has positive coefficients for ( 3 + 1 ) -free posets P.

B -symmetric chromatic function Vassiliev knot invariants. Chord diagrams. Sergei Algebra of chord diagrams. Chmutov A n is a C -vector space spanned by chord diagrams modulo four term relations: Stanley’s chromatic − + − = 0 . symmetric function. Vassiliev knot The vector space A := � invariants. A n has a natural bialgebra structure. Signed graphs n � 0 Weighted signed := = . Multiplication: × chromatic function. Bialgebra of � doubly Comultiplication: δ : A n → A k ⊗ A l is defined on chord diagrams by weighted k + l = n signed the sum of all ways to split the set of chords into two disjoint parts: graphs. � Open δ ( D ) := D J ⊗ D J . problems. J ⊆ [ D ] Primitive space P ( A ) is the space of elements D ∈ A with the property δ ( D ) = 1 ⊗ D + D ⊗ 1 . P ( A ) is also a graded vector space P ( A ) = � P n . n ≥ 1

B -symmetric chromatic function Vassiliev invariants. Bialgebra structure. Sergei Chmutov Stanley’s chromatic symmetric function. The classical Milnor—Moore theorem: any commutative and cocommutative Vassiliev knot bialgebra A is isomorphic to the symmetric tensor algebra of the primitive space, invariants. A ∼ = S ( P ( A )) . Signed graphs Weighted Let p 1 , p 2 , . . . be a basis for the primitive space P ( A ) then any element of A can signed be uniquely represented as a polynomial in commuting variables p 1 , p 2 , . . . . chromatic The dimensions of P n : function. Bialgebra of n 1 2 3 4 5 6 7 8 9 10 11 12 doubly weighted dim P n 1 1 1 2 3 5 8 12 18 27 39 55 signed graphs. Open problems.

B -symmetric chromatic function Vassiliev invariants. Weighted graphs. Sergei Chmutov S. Chmutov, S. Duzhin, S. Lando, Vassiliev knot invariants III. Forest algebra and Stanley’s chromatic weighted graphs , Advances in Soviet Mathematics 21 135–145 (1994). symmetric 1 1 6 5 function. 5 2 Vassiliev knot � � � � � � � � 2 2 invariants. 4 � � 3 3 � � Signed graphs 3 4 4 � � � � � � � � 6 Weighted 1 � � � � 6 signed 5 chromatic A chord diagram The intersection graph function. Bialgebra of doubly weighted signed Definition. A weighted graph is a graph G without loops and multiple edges given graphs. together with a weight w : V ( G ) → N that assigns a positive integer to each vertex Open of the graph. problems. Ordinary simple graphs can be treated as weighted graphs with the weights of all vertices equal to 1.

B -symmetric chromatic function Bialgebra of weighted graphs. Sergei Chmutov Stanley’s chromatic Let H n be a vector space spanned by all weighted graphs of the total weight n symmetric modulo the weighted contraction/deletion relation G = ( G \ e ) + ( G / e ) , where the function. graph G \ e is obtained from G by removing the edge e and G / e is obtained from Vassiliev knot G by a contraction of e such that if a multiple edge arises, it is reduced to a single invariants. edge and the weight w ( v ) of the new vertex v is set up to be equal to the sum of Signed graphs the weights of the two ends of the edge e . Weighted signed H := H 0 ⊕ H 1 ⊕ H 2 ⊕ . . . chromatic function. Multiplication: disjoint union of graphs; Bialgebra of doubly Comultiplication: splitting the vertex set into two subsets. weighted The primitive space P ( H n ) is of dimension 1 and spanned by a single vertex of signed weight n . graphs. The bialgebra H has a one-dimensional primitive space in each grading and thus Open problems. is isomorphic to C [ q 1 , q 2 , . . . ] .

B -symmetric chromatic function Weighted chromatic polynomial. Sergei Chmutov The image of an ordinary graph G (considered as a weighted graph with weights of all vertices equal to 1) in H can be represented by a polynomial W G ( q 1 , q 2 , . . . ) Stanley’s in the variables q n . chromatic symmetric function. S. Noble, D. Welsh, A weighted graph polynomial from chromatic invariants of Vassiliev knot knots , Annales de l’institut Fourier 49 (3) 1057–1087 (1999): invariants. � � ( − 1 ) | V ( G ) | W G = X G ( p 1 , p 2 , ... ) . Signed graphs � qj = − pj Weighted signed = q 2 Examples. W = ( • • ) + • 1 + q 2 chromatic 2 function. Bialgebra of doubly W = ( ) + = ( ) + 2 ( • • 2 ) + ( • 3 ) weighted 2 signed q 3 = 1 + 2 q 1 q 2 + q 3 graphs. ∞ � Open x m Plugging in q m = − p m = into the weighted chromatic polynomial a a graph problems. i i = 1 G with weight function w : V ( G ) → N we get the weighted chromatic function � � x w ( v ) X G , w ( x 1 , x 2 , . . . ) := κ ( v ) . κ : V ( G ) → N v ∈ V ( G ) proper