Exactly 14 intrinsically
knotted graphs have
21 edges.
Min Jung Lee,
jointwork with
Hyoung Jun Kim,
Hwa Jeong Lee
and Seungsang Oh
Contents
1. Definitions
2. Some results for intrinsically knotted
3. Terminology
4. Main theorem and lemmas
5. Sketch of proof
Definitions
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 contracting 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.
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Definitions
-The △-Y move ;
If there is △abc such that connection between vertices a, b, c, then it can be c
hanged by adding one vertex d and connecting d to all vertices a, b, c.
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Some results for IK
• [Conway-Gordon ]
Every embedding of K7 contains
a knotted cycle. (So, K7 is IK.)
• [Robertson-Seymour]
There is finite minor minimal graph for intrinsic knottedness.
- 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 knottedness.
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Some results for IK
• △-Y move preserve intrinsic knottedness.
Moreover, △-Y move preserve minor minimality
of K7 and K3,3,1,1, so thirteen graphs obtained
from K7 by △-Y move and twenty-five graphs
obtained from K3,3,1,1 by △-Y move are also
minor minimal for intrinsic knottedness.
From now on, we will consider about
triangle-free graph.
• [Goldberg, Mattman, and Naimi]
None of the six new graphs are intrinsically knotted.
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Some results for IK
• [Johnson, Kidwell, and Michael]
There is no intrinsically knotted graph
consisting at most 20 edges.
Main theorem
• The only triangle-free intrinsically
knotted graphs with 21 edges are
H12 and C14 .
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Terminology
• G=(E, V) : Simple triangle-free graph with deg(v) ≥ 3 for every vertex v in G.
^ ^ V)
^ : A graph obtained by removing 2 vertices and contracting edges which
• G=(E,
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.
^ = 21-|E(a)∪E(b)| - {|V (a)|+|V (b)|-|V (a, b)|+|V (a, b)|+|V (a, b)|}
|E|
3
3
3
4
Y
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Terminology
We can obtain the below equation easily ;
^ = 21-|E(a)∪E(b)| - {|V3(a)|+|V3(b)|-|V3(a, b)|+|V4(a, b)|+|VY(a, b)|}
|E|
b
a
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Main theorem and lemmas
A graph is n-apex if one can remove n vertices from it to obtain a planar graph.
Lemma 1. If G is a 2-apex, then G is not IK.
^ ≤ 8, then G^ is planar graph.
Lemma 2. If |E|
^ = 9, then G^ is planar graph, or homeomorpic to K3,3
Lemma 3. If |E|
Main theorem
• The only triangle-free intrinsically knotted graphs with 21 edges are
H12 and C14 .
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Sketch of proof
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 produce 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.
^ ≤ 21-(5+4)-{3+1}=8
|E|
b
a
b
^ ≤ 21-(5+4-1)-{3+3} ≤8
|E|
^ = 21-(5+5-1)-{3} =9
|E|
^ = 21-|E(a)∪E(b)| - {|V (a)|+|V (b)|-|V (a, b)|+|V (a, b)|+|V (a, b)|}
|E|
3
3
3
4
Y
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Sketch of proof
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.
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Thank you