Graphical neighborhoods of spatial graphs (Q2058963)

In classical knot theory, extensive use is made of the fact that a knot in the 3-sphere can be thickened to a tubular neighborhood and, for many purposes, the properties of the tubular neighborhood are interchangeable with those of the knot. Furthermore, it does not matter if one works in the piecewise-linear setting or the smooth setting. In this paper, the authors develop the analogous theory for spatial graphs in the smooth setting. A ``graphical neighborhood'' is the spatial graph version of the tubular neighborhood of a knot. Although spatial graphs have been extensively studied, previously no one has carefully developed the foundational topological theory. The main results of this paper include careful definitions of spatial graph and graphical neighborhood, the existence and uniqueness (up to a certain equivalence) of graphical neighborhoods, and the fact that \(S^3 \setminus N(G)\) is a deformation retract \(S^3 \setminus G\), where \(G\) is a spatial graph and \(N(G)\) is a graphical neighborhood. The main challenge of the proofs of each theorem lies in how vertices of degree 3 or greater are treated, as differential topology is classically concerned with manifolds and spatial graphs are not manifolds. For the entire collection see [Zbl 1459.37002].