Arc index of spatial graphs - Waseda University · 2016. 8. 7. · Arc index of spatial graphs...
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Arc index of spatial graphs
Sungjong No
jointwork with Minjung Lee and Seungsang Oh
International Workshop on Spatial Graphs 2016
August 5, 2016
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Introduction Examples Proof of Main Theorem
Contents
1 Introduction
2 Examples
3 Proof of Main Theorem
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Introduction Examples Proof of Main Theorem
Definition
• An arc presentation of a spatial graph G is an ambientisotopic image of G contained in the union of finitely manyhalf planes, called pages, with a common boundary line,called binding axis in such a way that each half planecontains a properly embedded single arc.
• An arc index α(G) of a graph G is a minimum of thenumber of arcs for arc presentations of a graph G.
=
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Introduction Examples Proof of Main Theorem
Definition
• To describe our results, decompose the spatial graph G intocut components by splitting and cutting along a maximalset of disjoint 2-spheres, each of which is either disjointfrom G or meeting G only at a vertex, while eachcomponent of their complement intersects with G.
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Introduction Examples Proof of Main Theorem
Definition
• To describe our results, decompose the spatial graph G intocut components by splitting and cutting along a maximalset of disjoint 2-spheres, each of which is either disjointfrom G or meeting G only at a vertex, while eachcomponent of their complement intersects with G.
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Introduction Examples Proof of Main Theorem
Definition
• A graph is called a bouquet if it is consist of one vertex andloops.
• If a cut-component is itself a bouquet spatial graph, we callit a bouquet cut-component .
bouquet bouquet cut-components
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Introduction Examples Proof of Main Theorem
Theorems for arc indices of knots
Bae-Park (2000)
Let K be an any nontrivial knot. Then α(K) ≤ c(K) + 2.Moreover if K is a non-alternating prime knot, thenα(K) ≤ c(K) + 1.
Jin-Park (2010)
Let K be an any non-alternating prime knot. Thenα(K) ≤ c(K).
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Introduction Examples Proof of Main Theorem
Theorem (Lee, No, Oh)
Let G be any spatial graph with e edges and b bouquetcut-components. Then
α(G) ≤ c(G) + e+ b.
Furthermore, this is the lowest possible upper bound.
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Introduction Examples Proof of Main Theorem
• trivial θn-curve
=
Naturally α(θn) = n (c(θn) = 0, e = n, b = 0)
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Introduction Examples Proof of Main Theorem
• 31 knot(regard it as a graph consist of one vertex and oneloop)
=
α(31) = 5 (c(31) = 3, e = 1, b = 1)
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Introduction Examples Proof of Main Theorem
• 51 θ-curve
=
• α(51) ≤ 8 (c(51) = 5, e = 3, b = 0)
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Introduction Examples Proof of Main Theorem
Wheel diagram and spoke
3
2
1
4
5
{1,4}
{2,5}
{1,4}
{3,5}
{2,4}
{1,3}
3
2
1
4
5{1,4} spoke
A wheel diagram is consist of α(G) spokes. To each spoke,assign the pair of numbers {a, b} when the arc corresponding tothe spoke connects the points numbered a and b on the binding.
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Introduction Examples Proof of Main Theorem
• Any spatial graph can be converted to a spoke diagram byspoking algorithm.
=
3
33
3
33
4 4 3
33
4 45
5 {3,5}3
3
4 45
{3,5}
3
3
44
{2,5}
2 3
3
4 4
{3,5}
{2,5} {2,6}
6
34
6{3,5}
{2,5} {2,6}
{3,7}
{4,7}
{3,5}
{2,5} {2,6}
{3,7}
{4,7}{1,4}
{1,3}
{1,6}
= 3
2
1
4
7
6
5
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Introduction Examples Proof of Main Theorem
• From now, we consider the diagram is of a cut component.
• Select a pivot vertex v0 of the diagram and we willconstruct a new diagram in three different ways accordingto the following three types;(Type 1) The vertex v comes from a crossing of Dm.(Type 2) The vertex v comes from a vertex of Dm which isnot v0.(Type 3) The pulling edge e is a loop, i.e., v = v0.
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Introduction Examples Proof of Main Theorem
(Type 1)
{1,4}
0v
3
1
i
j
ve e `
2e `
1e `0v
i
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e `2e `
1e `
kk
0v
i e `2e `
1e `
k
j k0v
i e `2e `
k
{j,k}
gnikopsgnidils
D `mD
m+1D
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Introduction Examples Proof of Main Theorem
(Type 2)
gnikopsgnidils 0v
e
2e
1e
4e
3e
kk
j k
ik k j `
{1,4}
0v
3
1
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ve
2e
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4e 3e j `
0v
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4e
3e
kkkkk
j `
0v
2e
3e
kk
{j,k}
{i,k}`{j ,k}
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Introduction Examples Proof of Main Theorem
(Type 3)
gnikopsgnidils
{1,4}
0v
3
1
i
j
e
0v
i
jk
k
0v j k
i k
{j,k}
{i,k}
0v
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Introduction Examples Proof of Main Theorem
• By spoking algorithm, the sum of spokes, regions andvertices is unchanged for Type 1 and 2 cases, while greaterthan by 1 for Type 3 case.
• If the diagram is a bouquet cut component, then Type 3case is necessary while the spoking algorithm is processing.
• We can adjust the spoking algorithm without increasingthe number of cut components.
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Introduction Examples Proof of Main Theorem
Example
3
33
3
33
4 4 {3,5}3
3
4 45
{3,5}
3
3
44
{2,5}
2 3
3
4 4
{3,5}
{2,5} {2,6}
6
34
6{3,5}
{2,5} {2,6}
{3,7}
{4,7}
{3,5}
{2,5} {2,6}
{3,7}
{4,7}{1,4}
{1,3}
{1,6}
regions : 8vertices :2spoke : 0
regions : 8vertices :2spoke : 0
regions : 7vertices :2spoke : 1
regions : 6vertices :2spoke : 2
regions : 5vertices :2spoke : 3
regions : 3vertices :2spoke : 5
regions : 1vertices :1spoke : 8
For every step, r(the number of regions)+s(the number ofspokes)+v(the number of vertices) is not changed.
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Introduction Examples Proof of Main Theorem
In the original diagram, c(G) + e+ 2 = r + v (s = 0). And inthe final diagram, r + v + s = s+ 2. So, c(G) + e = s. But finals is the number of arcs of this diagram. So, α(G) ≤ c(G) + e.
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Introduction Examples Proof of Main Theorem
Example of Type 3
regions : 4vertices :1spoke : 5
regions : 2 vertices :1spoke : 7
regions : 1vertices :1spoke : 9
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Introduction Examples Proof of Main Theorem
G1 G2 GGlue or combine cut-components at the intersecting vertex likethis figure. Then we get the inequality α(G) ≤ c(G) + e+ b.The proof is completed for the original graph G.
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Introduction Examples Proof of Main Theorem
Thank you
Arc index of spatial graphs Sungjong No