On Alexander Polynomials of Graphs Zhongtao Wu (joint with Yuanyuan - PowerPoint PPT Presentation

On Alexander Polynomials of Graphs Zhongtao Wu (joint with Yuanyuan Bao) The Chinese University of Hong Kong June 15, 2018 Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 1 / 20

Alexander polynomials The Alexander polynomial of links was first studied by J. W. Alexander in 1920s, which is a very useful and powerful invariant. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 2 / 20

Alexander polynomials The Alexander polynomial of links was first studied by J. W. Alexander in 1920s, which is a very useful and powerful invariant. Later people found many different ways of definitions: Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 2 / 20

Alexander polynomials The Alexander polynomial of links was first studied by J. W. Alexander in 1920s, which is a very useful and powerful invariant. Later people found many different ways of definitions: universal abelian cover of link complements Seifert surfaces Kauffman’s state sum formula Conway’s skein relations and many more ... Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 2 / 20

Spatial graph theory We study embedded graphs in this talk. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 3 / 20

Spatial graph theory We study embedded graphs in this talk. A graph is a finite collection of vertices V together with disjoint edges E connecting pairs of vertices. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 3 / 20

Spatial graph theory We study embedded graphs in this talk. A graph is a finite collection of vertices V together with disjoint edges E connecting pairs of vertices. An embedded graph G means a graph that exists in a specific position in the three-space, whereas an abstract graph g is a graph that is considered to be independent of any particular embedding. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 3 / 20

Spatial graph theory We study embedded graphs in this talk. A graph is a finite collection of vertices V together with disjoint edges E connecting pairs of vertices. An embedded graph G means a graph that exists in a specific position in the three-space, whereas an abstract graph g is a graph that is considered to be independent of any particular embedding. Spatial graph theory is the study of embedded graphs in the space. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 3 / 20

MOY graphs An MOY graph is an embedded graph equipped with a transverse orientation ... v L v ... a balanced coloring c : E → Z ≥ 0 such that for each vertex v ∈ V , � � c ( e ) = c ( e ) . incoming outgoing Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 4 / 20

MOY graphs A closed MOY graph is an MOY graph without vertex of valence one. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 5 / 20

MOY graphs A closed MOY graph is an MOY graph without vertex of valence one. Example (Singular) knots/links Embedded Θ graphs Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 5 / 20

Alexander polynomials of spatial graphs One can define an Alexander polynomial ∆ ( G , c ) ( t ) ∈ Z [ t ] / ± t n for closed MOY graphs via the following standard method in covering space. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 6 / 20

Alexander polynomials of spatial graphs One can define an Alexander polynomial ∆ ( G , c ) ( t ) ∈ Z [ t ] / ± t n for closed MOY graphs via the following standard method in covering space. coloring c determines a homomorphism φ c : π 1 ( S 3 − G ) → H 1 ( S 3 − G ; Z ) → Z � t � . Denote X = S 3 − G , and let ∂ in X be a subsurface of ∂ X . Let ˜ X be the cyclic covering of X corresponding to ker ( φ c ). The deck transformation endows the relative homology H 1 ( ˜ X , p − 1 ( ∂ in X )) with a Z [ t − 1 , t ]-module structure. Call this the Alexander module associated to ( G , c ). Define ∆ ( G , c ) ( t ) to be the determinant of a presentation matrix of the Alexander module. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 6 / 20

A second definition via Kauffman state sum Alternatively, ∆ ( G , c ) ( t ) can be defined in a more combinatorial and concrete manner. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 7 / 20

A second definition via Kauffman state sum Alternatively, ∆ ( G , c ) ( t ) can be defined in a more combinatorial and concrete manner. Starting with a graph projection/diagram D of G , draw a circle around each vertex. Cr ( D ): set of crossings Re ( D ): set of regions Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 7 / 20

A second definition via Kauffman state sum Alternatively, ∆ ( G , c ) ( t ) can be defined in a more combinatorial and concrete manner. Starting with a graph projection/diagram D of G , draw a circle around each vertex. Cr ( D ): set of crossings Re ( D ): set of regions Lemma | Re ( D ) | = | Cr ( D ) | + 2 if D is a connected graph diagram. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 7 / 20

Procedure of computing Kauffman state sum Mark a point δ on an edge and the two nearby regions. A state is a bijective map s : Cr ( D ) → Re ( D ) \{ marked regions } . Define local contributions M △ and A △ c . c -1 -1 1 1 1 1 ... ... 1 1 1 1 1 i i t i -i t t i/2 - t -i/2 t i -i 1 t 1 ... ... t -i/2 t i/2 1 1 i Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 8 / 20

Procedure of computing Kauffman state sum Sum over all states M s ( c ) A s ( c ) � � c c s crossing c Multiply the above sum with (a factor depending on δ ) and (another factor depending on D ) to get a graph invariant. Theorem The above invariant coincides with the Alexander polynomial ∆ ( G , c ) ( t ) defined earlier using cyclic covering; furthermore, it resolves the ± t n ambiguity and gives a normalized Alexander polynomial. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 9 / 20

An example of θ -curve The 5 1 in Litherland’s table of θ -curve diagrams j i i+j d d 1 2 e a 2 d b 1 d e 2 3 4 b a b 1 2 3 1 c c 2 1 δ * Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 10 / 20

An example of θ -curve The 5 1 in Litherland’s table of θ -curve diagrams j i i+j d d 1 2 e a 2 d b 1 d e 2 3 4 b a b 1 2 3 1 c c 2 1 δ * There are Kauffman states: a 1 b 1 c 1 d 2 e 1 , a 1 b 1 c 1 d 3 e 2 , a 1 b 1 c 2 d 4 e 2 , a 2 b 2 c 1 d 2 e 1 , a 2 b 2 c 1 d 3 e 2 , a 2 b 2 c 2 d 4 e 2 , a 2 b 3 c 2 d 1 e 2 Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 10 / 20

Calculations of the Alexander polynomial Compute each term of the state sum M s ( c ) A s ( c ) � � . c c s crossing c For example, the contribution of the state a 1 b 1 c 1 d 2 e 1 is 3 i 2 +2 j · ( t i + j 2 − t − i + j 2 − t − j j 2 ) · ( t 2 ) t Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 11 / 20

Calculations of the Alexander polynomial Compute each term of the state sum M s ( c ) A s ( c ) � � . c c s crossing c For example, the contribution of the state a 1 b 1 c 1 d 2 e 1 is 3 i 2 +2 j · ( t i + j 2 − t − i + j 2 − t − j j 2 ) · ( t 2 ) t After summing up and multiplying with the appropriate factors, we obtain the normalized Alexander polynomial 3 i +3 j 3 i + j i +3 j i + j i − j j − i − i − j 2 − t 2 + t 2 + t 2 + t 2 − t 2 ) · [ i + j ] ∆ ( G , c ) ( t ) = ( t − t 2 where [ i + j ] := t ( i + j ) / 2 − t − ( i + j ) / 2 t 1 / 2 − t − 1 / 2 Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 11 / 20

MOY graphic calculus Murakami, Ohtsuki and Yamada developed a graphic calculus for U q ( sl N )-polynomial invariants in the late 1990s. Defined for N ≥ 2 N = 2 case: Jones polynomial N = 0 case: Alexander polynomials Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 12 / 20