Open Access
June, 2000 On Transversally Simple Knots
Joan S. Birman, Nancy C. Wrinkle
J. Differential Geom. 55(2): 325-354 (June, 2000). DOI: 10.4310/jdg/1090340880


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 $\mathcal{TK}$ is transversally simple if it is determined by its topological knot type $\mathcal{K}$ and its Bennequin number. The main theorem asserts that any $\mathcal{TK}$ 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 [10]. As a second application we use a new theorem of Menasco [17] to extend a result of Etnyre [11] 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 $\mathcal{TK}$ is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.


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Joan S. Birman. Nancy C. Wrinkle. "On Transversally Simple Knots." J. Differential Geom. 55 (2) 325 - 354, June, 2000.


Published: June, 2000
First available in Project Euclid: 20 July 2004

zbMATH: 1026.57005
MathSciNet: MR1847313
Digital Object Identifier: 10.4310/jdg/1090340880

Rights: Copyright © 2000 Lehigh University

Vol.55 • No. 2 • June, 2000
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