On graphs for which every planar immersion lifts to a knotted spatial embedding

Abstract

We call a graph $G$ intrinsically linkable if there is a way to assign over/under information to any planar immersion of $G$ such that the associated spatial embedding contains a pair of nonsplittably linked cycles. We define intrinsically knottable graphs analogously. We show there exist intrinsically linkable graphs that are not intrinsically linked. (Recall a graph is intrinsically linked if it contains a pair of nonsplittably linked cycles in every spatial embedding.) We also show there are intrinsically knottable graphs that are not intrinsically knotted. In addition, we demonstrate that the property of being intrinsically linkable (knottable) is not preserved by vertex expansion.