A newly recognized intrinsically knotted graph

Abstract

In [Adams, 1994; The Knot Book], Colin Adams states as an open question whether removing a vertex and all edges incident to that vertex from an intrinsically knotted graph must yield an intrinsically linked graph. In this paper, we exhibit an intrinsically knotted graph for which there is a vertex that can be removed, and the resulting graph is not intrinsically linked. We further show that this graph is minor minimal with respect to being intrinsically knotted. ß

More intrinsically knotted graphs

“…In this case, C 3 could be (1, 2, 3), (1, 2, 4), (2, 3, 4), or (2,3,5). If C 3 is (1, 2, 3), then C 1 is (v, 4, 6, 5), and C 3 connects to 7 via edge (1, 7), which is disjoint from C 1 .…”

“…C 3 uses none of the vertices in {2, 6, 7}. Here C 3 must be (1,3,4) and C 1 must be (v, 2, 5). Here, C 3 connects to 6 via (4, 6) and to 7 via (1, 7).…”

“…Furthermore, there are exactly two sets of 4 vertices in H whose removal will disconnect H. Those two sets are {v, 2, 4, 6} and {v, 7, 9, 11}. The graph H − {v, 2, 4, 6} has four components; one is K 3,3 , and the other three components are all points. The graph H − {v, 7, 9, 11} has four components that are also K 3,3 and three points.…”

“…We will show our graphs are intrinsically knotted by showing that every embedding of them contains a subgraph that contracts onto an embedding of the graph D 4 with lk(C 1 , C 3 ) = 0 and lk(C 2 , C 4 ) = 0. This technique was used in [2] and [3].We will also make use of the following elementary lemma, which we state without proof. 6-cycle (a, 1, b, 2, c, 3), any two vertices in the set {a, b, c} are connected to any two vertices in the set {1, 2, 3} by disjoint edges.…”

“…We will show our graphs are intrinsically knotted by showing that every embedding of them contains a subgraph that contracts onto an embedding of the graph D 4 with lk(C 1 , C 3 ) = 0 and lk(C 2 , C 4 ) = 0. This technique was used in [2] and [3].…”