On graphs for which every planar immersion lifts to a knotted spatial embedding. (English) Zbl 1154.57003

Decelles, Amy; Foisy, Joel; Versace, Chad; Wilson, Alice

Involve 1, No. 2, 145-158 (2008).

A graph is intrinsically linked (respectively knotted) if, for every planar immersion of the graph, 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 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.

Reviewer: Iain Moffatt (Mobile, AL)

MSC:

57M15	Relations of low-dimensional topology with graph theory
57M25	Knots and links in the \(3\)-sphere (MSC2010)
05C10	Planar graphs; geometric and topological aspects of graph theory

Keywords:

spatial graphs; intrinsically linkable; intrinsically knottable; intrinsically linked; intrinsically knotted; vertex expansion