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 PresentationTRANSCRIPT
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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
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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 .
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• 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
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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
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Thank you