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

1. Definitions 2. Some results for intrinsically knotted 3. Terminology 4. Main theorem and lemmas 5. Sketch of proof Contents

3/ 12 We will consider a graph as an embedded graph in R 3. -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. Definitions

4/ 12 -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. Definitions

5/ 12 [Conway-Gordon ] Every embedding of K 7 contains a knotted cycle. (So, K 7 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. - K 7 and K 3,3,1,1 are minor minimal graphs for intrinsic knottedness. Some results for IK

6/ 12 △ -Y move preserve intrinsic knottedness. Moreover, △ -Y move preserve minor minimality of K 7 and K 3,3,1,1, so thirteen graphs obtained from K 7 by △ -Y move and twenty-five graphs obtained from K 3,3,1,1 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.

7/ 12 [Johnson, Kidwell, and Michael] There is no intrinsically knotted graph consisting at most 20 edges. Some results for IK The only triangle-free intrinsically knotted graphs with 21 edges are H12 and C 14. Main theorem

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.  V n (a) : The set of neighboring vertices of a with degree n.  V n (a, b) = V n (a) ∩ V n (b).  V Y (a, b) : The set of vertices of V 3 (a, b) whose neighboring vertices are a, b and a vertex with degree 3. 8 / 12 Terminology ^^^ |E| = 21-|E(a) ∪ E(b)| - {|V 3 (a)|+|V 3 (b)|-|V 3 (a, b)|+|V 4 (a, b)|+|V Y (a, b)|} ^

9 / 12 Terminology We can obtain the below equation easily ; |E| = 21-|E(a) ∪ E(b)| - {|V 3 (a)|+|V 3 (b)|-|V 3 (a, b)|+|V 4 (a, b)|+|V Y (a, b)|} a ^ b

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 K 3,3 10 / 12 Main theorem and lemmas The only triangle-free intrinsically knotted graphs with 21 edges are H12 and C 14. Main theorem ^^ ^^

11 / 12 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. Sketch of proof a b b |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)| - {|V 3 (a)|+|V 3 (b)|-|V 3 (a, b)|+|V 4 (a, b)|+|V Y (a, b)|} ^

12 / 12 When deg(a) = 4, it is enough to consider three cases (|V 3 |, |V 4 |) = (2, 9) or (6,6) or (10, 3) where |V n | 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 H 12. 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 C 14. This is end of the proof. Sketch of proof

Thank you