Knots, braid index and dynamical type
We recall first a couple of well-known facts. Given a knot \(K\) in the 3- sphere \(S^3\), a theorem of Thurston says that \(K\) is either a torus knot, a satellite knot, or its complement \(S^3 - K\) admits a hyperbolic structure. In the latter case, the knot is said to be hyperbolic. The braid index set \(Bi (K)\) of \(K\) is the set of braids representing \(K\) with the minimum number of strands. A braid on \(n\) strands defines an isotopy class of homeomorphisms of the \(n\)-punctured disk. Homeomorphisms of surfaces are classified, by another result of Thurston, into three types: periodic, reducible and pseudo-Anosov. In this paper, the author studies the relation between the two results of Thurston and he proves the following: (i) \(K\) is a torus knot if and only if every braid in \(Bi (K)\) is periodic. (ii) \(K\) is a hyperbolic or a non-braided satellite knot if and only if every braid in \(Bi (K)\) is pseudo-Anosov; (iii) \(K\) is a braided satellite knot if and only if for every \(\beta\) in \(Bi (K)\), there exists a finite sequence of exchange moves (and conjugacies) in \(Bi (K)\): \(\beta \to \beta_1 \to \cdots \to \beta_N\) such that \(\beta_N\) is reducible. (Exchange moves are certain transformations which have been defined by Birman and Menasco.) Another natural idea which the author exploits is the definition of an invariant for a knot \(K\) arising from the dynamics of the homeomorphisms of the punctured disk defined by the braids in \(Bi(K)\). The author introduces an invariant which he denotes by \(\Lambda_K\) and which has the following properties: (i) \(\Lambda_K = 1\) if and only if \(K\) is a torus knot or an iterated torus knot; (i) If \(K\) has braid index \(n\) and is not a torus knot or an iterated torus knot then \(\Lambda_K \geq 2^{1/n}\).
scientific article; zbMATH DE number 590277