On the total curvature of semialgebraic graphs

TL;DR

This paper introduces a new notion of total curvature for semialgebraic graphs in 3D space, establishing inequalities and characterizing when graphs are knotted based on their curvature using Morse theory.

Contribution

It defines total curvature for semialgebraic graphs, proves a Chern-Lashof type inequality, and generalizes classical knotting results with Morse theoretic methods.

Findings

Total curvature satisfies a Chern-Lashof type inequality.

Equality cases are characterized precisely.

Certain graphs cannot be knotted if their curvature is below a threshold.

Abstract

We define the total curvature of a semialgebraic embedding of a graph in the 3-dimensional Euclidean space. We prove that it satisfies a Chern-Lashof type inequality and we describe when the equality holds. We also prove a generalization of a classical result of Fary and Milnor stating that certain graphs cannot be knotted if they are not too curved. The techniques employed are Morse theoretic.