Total curvature for knotted surfaces
This paper deals with the total absolute curvature \(\tau\) of submanifolds in the case that the submanifolds are knotted. Previous work by Fary, Milnor, Langevin, Morton and Wintgen is improved and generalized. The first theorem says that for any imbedding \(f: M^ n\to E^ N\), \(n=2\) or \(n\geq 4\), there holds the inequality \(\tau (f)\geq \beta (M)+4\sigma_ 1\) where \(\beta\) (M) is the sum of the Betti numbers with coefficients in \({\mathbb{Z}}_ 2\) and \(\sigma_ 1\) denotes the first homotopy excess. \(\sigma_ 1\) is zero if f is unknotted and it counts the number of necessary additional generators for the fundamental group of the complement (the sum of both if the complement is not connected). For \(n=1\) one has \(\tau \geq \beta +2\sigma_ 1\) and for \(n=3\) the weaker inequality by \textit{P. Wintgen} [Beitr. Algebra Geom. 10, 87-96 (1980; Zbl 0472.53057)] could not be improved. Then the notions of isotopy tightness and relative isotopy tightness are introduced. Isotopy tight means that \(\tau\) attains its minimal value in a given isotopy class, and the relative isotopy tightness is a ''local'' version where only nearby imbeddings are considered. For surfaces of genus \(g\leq 2\) no isotopy tight imbeddings could be found but for any genus \(g\geq 3\) there are examples derived from a ''key example'' of genus 3 with \(\tau =12\) and \(\sigma_ 1=1\). By considering a tubular neighbourhood of this key example the authors are able to construct a hypersurface with \(\beta =8\) and \(\tau =12\) where this total absolute curvature is minimal possible. The last part of the paper deals with relatively isotopy tight surfaces. A complete classification of such RIT-surfaces up to genus two and \(\tau =10\) is made plausible and stated in a Pre-theorem. It says that there is one type for \(g=1\) and \(\tau =8\) and four types for \(g=2\) and \(\tau =10\).
scientific article; zbMATH DE number 3881436
scientific article