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A new knot invariant. (English) Zbl 0632.57006
The author introduces a new knot invariant, based on Schubert’s bridge number, but which captures more of the complexity of the knot. This invariant is defined as follows. Given a smooth closed curve K embedded in \({\mathbb{R}}^ 3\) and a line \(\delta\) in \({\mathbb{R}}^ 3\), the bridge number of K with respect to \(\delta\) is the number of components of the critical set of the orthogonal projection from K to \(\delta\). The author defines the superbridge number of K as the maximum of this bridge number when \(\delta\) ranges over all directions in \({\mathbb{R}}^ 3\). And the superbridge index of the knot [K] is the minimum of the superbridge number over all curves isotopic to K. The main result of the article is an explicit computation of the superbridge index of torus knots.
Reviewer: F.Bonahon

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:
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