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Exactly 14 intrinsically knotted graphs have 21 edges. Min Jung Lee, jointwork with Hyoung Jun Ki Hwa Jeong Lee and Seungsang Oh

Exactly 14 intrinsically knotted graphs have 21 edges

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Exactly 14 intrinsically knotted graphs have 21 edges . Min Jung Lee, jointwork with Hyoung Jun Kim, Hwa Jeong Lee and Seungsang Oh. Contents. Definitions Some results for intrinsically knotted Terminology Main theorem and lemmas Sketch of proof. - PowerPoint PPT Presentation

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Page 1: Exactly 14 intrinsically    knotted graphs have                     21 edges

Exactly 14 intrinsically knotted graphs have 21 edges.

Min Jung Lee,

jointwork with Hyoung Jun Kim, Hwa Jeong Lee and Seungsang Oh

Page 2: Exactly 14 intrinsically    knotted graphs have                     21 edges

1.Definitions

2.Some results for intrinsically knotted

3.Terminology

4.Main theorem and lemmas

5.Sketch of proof

Contents

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We will consider a graph as an embedded graph in R3.

-A graph G is called intrinsically knotted (IK) if every spatial embedding of the graph contains a knotted cycle.

-For a graph G, H is minor graph of G obtained by edge con-tracting or edge deleting from G.

-If no minor graph of G are intrinsically knotted even if G is intrinsically knotted , G is called minor minimal for intrinsic knottedness.

Definitions

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-The △-Y move ;

If there is △abc such that connection between vertices a, b, c, then it can be changed by adding one vertex d and con-necting d to all vertices a, b, c.

Definitions

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• [Conway-Gordon ] Every embedding of K7 contains a knotted cycle. (So, K7 is IK.)

• [Robertson-Seymour] There is finite minor minimal graph for intrinsic knotted-ness.

- But completing the set of minor minimal for intrinsic knottedness is still open problem.

- K7 and K3,3,1,1 are minor minimal graphs for intrinsic knot-tedness.

Some results for IK

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• △-Y move preserve intrinsic knotted-ness.

Moreover, △-Y move preserve minor minimalityof K7 and K3,3,1,1, so thirteen graphs ob-tainedfrom K7 by △-Y move and twenty-five graphs obtained from K3,3,1,1 by △-Y move are also minor minimal for intrinsic knottedness.

Some results for IK

• [Goldberg, Mattman, and Naimi] None of the six new graphs are intrinsically knotted.

From now on, we will consider about triangle-free graph.

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• [Johnson, Kidwell, and Michael] There is no intrinsically knot-ted graph consisting at most 20 edges.

Some results for IK

Main theorem

• The only triangle-free intrinsically knotted graphs with 21 edges are H12 and C14 .

Page 8: Exactly 14 intrinsically    knotted graphs have                     21 edges

• G=(E, V) : Simple triangle-free graph with deg(v) ≥ 3 for ev-ery vertex v in G.

• G=(E, V) : A graph obtained by removing 2 vertices and con-tracting edges which have degree 1 or 2 vertex at either end.

E(a) : The set of edges which are incident with a. V(a) : The set of neighboring vertices of a. Vn(a) : The set of neighboring vertices of a with degree n.

Vn(a, b) = Vn(a) ∩ Vn(b). VY(a, b) : The set of vertices of V3(a, b) whose neighboring vertices are a, b and a vertex with degree 3.

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Terminology

^ ^ ^

|E| = 21-|E(a)∪E(b)| - {|V3(a)|+|V3(b)|-|V3(a, b)|+|V4(a, b)|+|VY(a, b)|}^

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Terminology

We can obtain the below equation easily ;

|E| = 21-|E(a)∪E(b)| - {|V3(a)|+|V3(b)|-|V3(a, b)|+|V4(a, b)|+|VY(a, b)|}

a

^

b

Page 10: Exactly 14 intrinsically    knotted graphs have                     21 edges

A graph is n-apex if one can remove n vertices from it to ob-tain a planar graph.

Lemma 1. If G is a 2-apex, then G is not IK.Lemma 2. If |E| ≤ 8, then G is planar graph.Lemma 3. If |E| = 9, then G is planar graph, or homeomorpic to K3,3

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Main theorem and lemmas

Main theorem• The only triangle-free intrinsically knotted graphs with 21

edges are H12 and C14 .

^ ^^ ^

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Let a be a vertex which has maximum degree in G = (V, E).

Our proof treats the cases deg(a) = 7, 6, 5, 4, 3 in turn. In most cases, we delete a vertex a and another vertex to pro-duce a planar graph. And we will consider subcase with the number of degree 3 vertex in each deg(a) = 7, 6, 5 case. In these cases, we show that the graph G is 2-apex, so G is not intrinsically knotted.

Sketch of proof

abb

|E| ≤ 21-(5+4)-{3+1}=8

|E| ≤ 21-(5+4-1)-{3+3} ≤8

|E| = 21-(5+5-1)-{3} =9

^

^

^

|E| = 21-|E(a)∪E(b)| - {|V3(a)|+|V3(b)|-|V3(a, b)|+|V4(a, b)|+|VY(a, b)|}^

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When deg(a) = 4, it is enough to consider three cases (|V3|, |V4|) = (2, 9) or (6,6) or (10, 3)

where |Vn| is the number of degree n vertex.

We show that the case (2, 9) and (10, 3) are not intrinsically knotted, and the case (6, 6) is homeomorphic to H12.

The last case is deg(a) = 3. So all vertex have degree 3. In this case, we can know that the graph is homeomorphic to C14.

This is end of the proof.

Sketch of proof

Page 13: Exactly 14 intrinsically    knotted graphs have                     21 edges

Thank you