Exactly 14 intrinsically knotted graphs have 21 edges .

1. Exactly 14 intrinsically knotted graphs have 21 edges. Min Jung Lee, jointwork with Hyoung Jun Kim, HwaJeong Lee and Seungsang Oh

2. Contents Definitions Some results for intrinsically knotted Terminology Main theorem and lemmas Sketch of proof

3. 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, His 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 minorminimal for intrinsic knottedness. 3/ 12

4. Definitions -The △-Y move ; If there is △abcsuch that connection between vertices a,b,c, then it can be changed by adding one vertex d and connecting d to all vertices a,b,c. 4/ 12

5. Some results for IK • [Conway-Gordon] • Every embedding of K7 contains • a knotted cycle. (So, K7is 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. 5/ 12

6. Some results for IK • △-Y move preserve intrinsic knottedness. • Moreover, △-Y move preserve minor minimality • of K7and K3,3,1,1, so thirteen graphs obtained • from K7by △-Y move and twenty-five graphs • obtained from K3,3,1,1by △-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. 6/ 12

7. Some results for IK • [Johnson, Kidwell, and Michael] • There is no intrinsically knotted graph • consisting at most 20 edges. 7/ 12

8. Terminology • G=(E, V) : Simple triangle-free graph with deg(v)≥3 for every vertex v in G. • G=(E,V) : A graph obtained by removing 2 vertices and contracting 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. ^ ^ ^ ^ • |E| = 21-|E(a)∪E(b)| - {|V3(a)|+|V3(b)|-|V3(a, b)|+|V4(a, b)|+|VY(a, b)|} 8 / 12

9. 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)|} ^ b a 9 / 12

10. 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. 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 ^ ^ ^ ^ 10 / 12

11. 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. ^ |E| ≤ 21-(5+4)-{3+1}=8 ^ |E| ≤ 21-(5+4-1)-{3+3} ≤8 b a b ^ |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)|} 11 / 12

12. 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. 12 / 12

13. Thank you