The total curvature of a knotted torus
In this paper dedicated to James Eells jun., the authors generalize to torus embeddings in \({\mathbb{R}}^ 3\) the theorem of Milnor: The normalized total curvature of an honest knot \(\gamma\) is \(\pi (\gamma)=\int | \rho ds| \quad /\quad \pi >2 B[\gamma].\) The infimum of \(\tau\) in the isotopy class [\(\gamma\) ] is twice the bridge index B[\(\gamma\) ] of the knot. A torus T, embedded in \({\mathbb{R}}^ 3\subset {\mathbb{R}}^ 3\cup \{\infty \}=S^ 3\), divides \(S^ 3\) in two parts, one of which is a tube around a knot \(\gamma\). Theorem: The normalized total curvature \(\tau (T)=\int | K d\sigma | /2\pi\) of an honestly knotted torus \(T\subset {\mathbb{R}}^ 3\) has infimum 4 B[\(\gamma\) ]. This infimum cannot be attained: \(\tau (T)>4 B[\gamma]\). This paper uses results of an earlier paper [Invent. Math. 77, 25-69 (1984; Zbl 0553.53034)] where the authors proved attainment of an infimum for certain knotted surfaces of genus \(g\geq 3\).
scientific article; zbMATH DE number 3971671