Journal of Differential Geometry

On Transversally Simple Knots

Joan S. Birman and Nancy C. Wrinkle

Full-text: Open access

Abstract

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.

Article information

Source
J. Differential Geom., Volume 55, Number 2 (2000), 325-354.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090340880

Digital Object Identifier
doi:10.4310/jdg/1090340880

Mathematical Reviews number (MathSciNet)
MR1847313

Zentralblatt MATH identifier
1026.57005

Citation

Birman, Joan S.; Wrinkle, Nancy C. On Transversally Simple Knots. J. Differential Geom. 55 (2000), no. 2, 325--354. doi:10.4310/jdg/1090340880. https://projecteuclid.org/euclid.jdg/1090340880


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