A Study of Intrinsic Knots and Links: a Tale of Two Graphs Alex Barylskiy and Hannah Schwartz

Abstract

Our research focused on two questions in graph and knot theory. Our first question addressed the complete graph of nine vertices (K9) and whether or not a 3-linkless straight-edge embedding is realizable in 3-space. A previously published paper provided a 3-linkless drawing of K9 with nonlinear edges. However we proved that this orientation is impossible because it contains a knotted cycle of 5 vertices, which is impossible in three dimensions. No other significantly different embeddings of K9 were found which eliminated all triple links. Secondly, we discovered an intrinsically knotted graph (G11,22) with only 5 members in its delta-wye family by adding an edge to a unknotted cousin of K7. We proved that this graph is minor minimal intrinsically knotted by finding unknotted embeddings for all nine minors of G11,22. This proved the existence of the smallest family of a MMIK graph yet to be discovered.