On graphs for which every planar immersion lifts to a knotted spatial embedding (Q940979)

A graph is intrinsically linked (respectively knotted) if, for every planar immersion of the graph, \textit{every} assignment of a crossing structure results in a diagram for a spatial graph that contains a pair of linked cycles (respectively a knotted cycle). A graph is intrinsically linkable (respectively knottable) if, for every planar immersion of the graph, there exists \textit{an} assignment of a crossing structure that results in a diagram for a spatial graph that contains a pair of linked cycles (respectively a knotted cycle). This paper is largely concerned with the distinction between intrinsically linkable (respectively knottable) and intrinsically linked (respectively knotted) graphs. In this paper the authors provide examples of intrinsically linkable graphs that are not intrinsically linked and examples of intrinsically knottable graphs that are not intrinsically knotted. These examples show that although an intrinsically linked (respectively knotted) graph is necessarily intrinsically linkable (respectively knottable), the converse is not true. Examples are also given to show that the property of a graph being intrinsically linkable (respectively knottable) is not preserved by vertex expansion.