Total curvature and isotopy of graphs in $R^3$

TL;DR

This paper introduces a notion of net total curvature for graphs in three-dimensional space and establishes bounds that determine when such graphs are isotopic to planar or convex configurations, extending classical knot theory results.

Contribution

It defines net total curvature for graphs in R^3 and proves bounds that relate curvature to isotopy classes, including extensions to continuous graphs.

Findings

Net total curvature of a theta-graph is at least 3π.

Graphs with net total curvature less than 4π are isotopic to planar theta-graphs.

Continuous graphs of finite total curvature are isotopic to polygonal graphs.

Abstract

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The F\'ary-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4\pi$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs $\Gamma$, what limitations on the isotopy class of $\Gamma$ are implied by a bound on total curvature? What does ``total curvature" mean for a graph? We define a natural notion of net total curvature of a graph $\Gamma$ in $R^3$, and prove that if $\Gamma$ is homeomorphic to the $\theta$-graph, then the net total curvature of $\Gamma$ \geq 3\pi$; and if it is $< 4\pi$, then $\Gamma$ is isotopic in $R^3$ to a planar $\theta$-graph. Further, the net total curvature $= 3\pi$ only when $\Gamma$ is a convex plane curve plus a chord. We begin our discussion with piecewise smooth graphs, and extend all these results to continuous graphs in the final section. In particular, we show that continuous graphs of finite total curvature are isotopic to polygonal graphs.