Banchoff, Thomas F. Osculating tubes and self-linking for curves on the three-sphere. (English) Zbl 1008.53005 Fernández, Marisa (ed.) et al., Global differential geometry: the mathematical legacy of Alfred Gray. Proceedings of the international congress on differential geometry held in memory of Professor Alfred Gray, Bilbao, Spain, September 18-23, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 288, 10-19 (2001). Author’s abstract: For a smooth closed curve, the osculating tube consists of the union of the osculating circles at points of the curve. For a closed curve on the three-sphere, the self-linking number of the image of the curve under stereographic projection from a point to the opposite hyperplane depends on the point, and the difference between the numbers for two points is related to the algebraic number of times that a curve from one point to the other meets the osculating tube of the curve transversely. The self-linking number of an \((n,m)\)-torus knot on the three-sphere is shown to be \(nm\), answering a question of N. Kuiper.For the entire collection see [Zbl 0980.00033]. Reviewer: T.Koufogiorgos (Ioannina) Cited in 2 Documents MSC: 53A04 Curves in Euclidean and related spaces 53A30 Conformal differential geometry (MSC2010) Keywords:self-linking number; stereographic projection; osculating circles; osculating tube PDFBibTeX XMLCite \textit{T. F. Banchoff}, Contemp. Math. 288, 10--19 (2001; Zbl 1008.53005)