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
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
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
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
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
We study embedded graphs in this talk.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 3 / 20
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
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
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
An MOY graph is an embedded graph equipped with a transverse orientation
a balanced coloring c : E → Z≥0 such that for each vertex v ∈ V ,
c(e) =
c(e).
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 4 / 20
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
A closed MOY graph is an MOY graph without vertex of valence one.
(Singular) knots/links Embedded Θ graphs
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 5 / 20
One can define an Alexander polynomial ∆(G,c)(t) ∈ Z[t]/ ± tn 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
One can define an Alexander polynomial ∆(G,c)(t) ∈ Z[t]/ ± tn for closed MOY graphs via the following standard method in covering space. coloring c determines a homomorphism φc : π1(S3 − G) → H1(S3 − G; Z) → Zt. Denote X = S3 − G, and let ∂inX be a subsurface of ∂X. Let ˜ X be the cyclic covering of X corresponding to ker(φc). The deck transformation endows the relative homology H1( ˜ X, p−1(∂inX)) 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
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
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
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
|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
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△
c
and A△
c .
1 1 1 1 1 1
1 1 1 1 1 1 1
i/2 i/2
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 8 / 20
Sum over all states
Ms(c)
c
As(c)
c
Multiply the above sum with (a factor depending on δ) and (another factor depending on D) to get a graph invariant.
The above invariant coincides with the Alexander polynomial ∆(G,c)(t) defined earlier using cyclic covering; furthermore, it resolves the ±tn ambiguity and gives a normalized Alexander polynomial.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 9 / 20
The 51 in Litherland’s table of θ-curve diagrams
i+j j i * δ e d c b a
4
e
1
d d d c b b a
1 1 1 1 2 2 2 2 2 3 3
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 10 / 20
The 51 in Litherland’s table of θ-curve diagrams
i+j j i * δ e d c b a
4
e
1
d d d c b b a
1 1 1 1 2 2 2 2 2 3 3
There are Kauffman states: a1b1c1d2e1, a1b1c1d3e2, a1b1c2d4e2, a2b2c1d2e1, a2b2c1d3e2, a2b2c2d4e2, a2b3c2d1e2
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 10 / 20
Compute each term of the state sum
Ms(c)
c
As(c)
c
. For example, the contribution of the state a1b1c1d2e1 is t
3i 2 +2j · (t i+j 2 − t− i+j 2 ) · (t j 2 − t− j 2 ) Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 11 / 20
Compute each term of the state sum
Ms(c)
c
As(c)
c
. For example, the contribution of the state a1b1c1d2e1 is t
3i 2 +2j · (t i+j 2 − t− i+j 2 ) · (t j 2 − t− j 2 )
After summing up and multiplying with the appropriate factors, we
∆(G,c)(t) = (t
3i+3j 2
− t
3i+j 2 − t i+3j 2 + t i+j 2 + t i−j 2 + t j−i 2 − t −i−j 2 ) · [i + j]
where [i + j] := t(i+j)/2−t−(i+j)/2
t1/2−t−1/2
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 11 / 20
Murakami, Ohtsuki and Yamada developed a graphic calculus for Uq(slN)-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
(Move 0)
= N i
⎛ ⎜ ⎜ ⎝ i i j + i j ⎞ ⎟ ⎟ ⎠
N
= N − i j
⎜ ⎜ ⎝ i ⎞ ⎟ ⎟ ⎠
N
(Move 2) ⎛ ⎜ ⎜ ⎝ i i i − j j ⎞ ⎟ ⎟ ⎠
N
=
j
⎜ ⎜ ⎝ i ⎞ ⎟ ⎟ ⎠
N
(Move 3) ⎛ ⎜ ⎜ ⎜ ⎝ i + j + k k i + j i j ⎞ ⎟ ⎟ ⎟ ⎠
N
= ⎛ ⎜ ⎜ ⎜ ⎝ i + j + k k j + k i j ⎞ ⎟ ⎟ ⎟ ⎠
N
(Move 4) ⎛ ⎜ ⎜ ⎜ ⎝ 1 i i 1 i 1 i + 1 i + 1 ⎞ ⎟ ⎟ ⎟ ⎠
N
= [N − i − 1] ⎛ ⎜ ⎜ ⎜ ⎝ 1 i i − 1 1 i ⎞ ⎟ ⎟ ⎟ ⎠
N
+ ⎛ ⎜ ⎜ ⎜ ⎝ 1 i ⎞ ⎟ ⎟ ⎟ ⎠
N
(Move 5) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 i + j − 1 i + k i i + k − 1 j − k k j ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
N
=
k − 1
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 i + j − 1 i j i + j ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
N
+
k
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ i 1 j i + j − 1 i − 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
N
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 13 / 20
⎛ ⎜ ⎜ ⎜ ⎝ i j ⎞ ⎟ ⎟ ⎟ ⎠
N
=
i
(−1)k+(j+1)iqi−k ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ j i j + k i k i − k j + k − i j ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
N
Figure 2. MOY resolutions of a coloured knot diagram if i ≤ j ⎛ ⎜ ⎜ ⎜ ⎝ i j ⎞ ⎟ ⎟ ⎟ ⎠
N
=
i
(−1)k+(i+1)jqj−k ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ j i j − k i k i + k i + k − j j ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
N
Figure 3. MOY resolutions of a coloured knot diagram if i > j
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 14 / 20
Using the above Kauffman state sum formulation, one can check that ∆(D,c)(t) satisfies a set of 10 relations.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 15 / 20
Using the above Kauffman state sum formulation, one can check that ∆(D,c)(t) satisfies a set of 10 relations. Conversely, these 10 relations uniquely determine ∆(D,c)(t) for all closed MOY graphs. Thus, it may be viewed as a third definition of Alexander polynomials.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 15 / 20
Using the above Kauffman state sum formulation, one can check that ∆(D,c)(t) satisfies a set of 10 relations. Conversely, these 10 relations uniquely determine ∆(D,c)(t) for all closed MOY graphs. Thus, it may be viewed as a third definition of Alexander polynomials. In particular, this gives one more interpretation of the classical Alexander polynomials of links.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 15 / 20
Symmetries: If G ∗ is the mirror image of G, then ∆(G,c)(t) = ∆(G ∗,c)(t−1). If −G is the graph with opposite orientation, then ∆(G,c)(t) = ∆(−G,c)(t).
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 16 / 20
Symmetries: If G ∗ is the mirror image of G, then ∆(G,c)(t) = ∆(G ∗,c)(t−1). If −G is the graph with opposite orientation, then ∆(G,c)(t) = ∆(−G,c)(t). Positivity: Suppose G is a planar graph of at least 1 vertex equipped with a non-negative coloring c. Then the non-zero coefficients of the Alexander polynomial ∆(G,c)(t) are all positive, that is, ∆(G,c)(t) ∈ Z≥0[t± 1
4 ]. Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 16 / 20
Non-vanishing: If G is a connected planar graph equipped with a positive coloring c, then the Alexander polynomial is non-vanishing: ∆(G,c)(t) = 0.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 17 / 20
Non-vanishing: If G is a connected planar graph equipped with a positive coloring c, then the Alexander polynomial is non-vanishing: ∆(G,c)(t) = 0. Intrinsic invariant: Suppose G and G ′ are two different spatial embedding of an abstract directed graph g, then the value of the Alexander polynomial evaluated at t = 1 is the same: ∆(G,c)(1) = ∆(G ′,c)(1). In other words, ∆(g,c) := ∆(G,c)(1) is an intrinsic invariant of an abstract graph g.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 17 / 20
In 2006, Viro defined an Alexander polynomial ∆(G, c) for framed graphs via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1).
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 18 / 20
In 2006, Viro defined an Alexander polynomial ∆(G, c) for framed graphs via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1). Bao observed that ∆(G, c) satisfies a set of relations that are nearly identical to the MOY relations for our Alexander polynomial ∆(G, c). Consequently, the two Alexander polynomial invariant are essentially the same.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 18 / 20
In a work in progress, we define a Heegaard Floer homology for MOY
following properties.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 19 / 20
In a work in progress, we define a Heegaard Floer homology for MOY
following properties. The homology HFG −
d (G, s) is bigraded with a Maslov grading d and
an Alexander grading s, both depending on the color c.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 19 / 20
In a work in progress, we define a Heegaard Floer homology for MOY
following properties. The homology HFG −
d (G, s) is bigraded with a Maslov grading d and
an Alexander grading s, both depending on the color c. For a planar graph G, the group HFG −(G) is determined by the Alexander polynomial ∆(G,c)(T); indeed, the homology HFG −
d (G, s)
is supported on the line 2s = d.
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 19 / 20
In a work in progress, we define a Heegaard Floer homology for MOY
following properties. The homology HFG −
d (G, s) is bigraded with a Maslov grading d and
an Alexander grading s, both depending on the color c. For a planar graph G, the group HFG −(G) is determined by the Alexander polynomial ∆(G,c)(T); indeed, the homology HFG −
d (G, s)
is supported on the line 2s = d. In general, the Euler characteristic gives the Alexander polynomial, in the sense that:
(−1)ddim(HFG −
d (G, s)) · ts = ∆(G,c)(t).
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 19 / 20
Zhongtao Wu (joint with Yuanyuan Bao) (CUHK) On Alexander Polynomials of Graphs June 15, 2018 20 / 20