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On transversally simple knots. (English) Zbl 1026.57005

Summary: This paper studies knots that are transversal to the standard contact structure in \(\mathbb{R}^3\), bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type \({\mathcal T}{\mathcal K}\) is transversally simple if it is determined by its topological knot type \({\mathcal K}\) and its Bennequin number. The main theorem asserts that any \({\mathcal T}{\mathcal K}\) whose associated \({\mathcal K}\) satisfies a condition that we call exchange reducibility is transversally simple. As a first application, we prove that the unlink is transversally simple, extending the main theorem in [Y. Eliashberg, Topological Methods in modern mathematics, 171-193, Publish or Perish, Inc. (1993; Zbl 0809.53033)]. As a second application we use a new theorem of Menasco [W. Menasco, Geom. Topol. 5, 651-682 (2001; Zbl 1002.57025)] to extend a result of Etnyre [J. Etnyre, Geom. Topol. 3, 253-268 (1999; Zbl 0927.57004)] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on \({\mathcal K}\) in order to prove that any associated \({\mathcal T}{\mathcal K}\) is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R17 Symplectic and contact topology in high or arbitrary dimension
57R52 Isotopy in differential topology
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