Celebratio Mathematica — Birman

My work with Joan Birman

Works connected to Nancy Court Wrinkle

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J. S. Birman and N. C. Wrinkle: “On transversally simple knots,” J. Diff. Geom. 55 : 2 (2000), pp. 325–354. MR 1847313 Zbl 1026.57005 article

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 [Eliashberg 1991]. As a second application we use a new theorem of Menasco [2001] to extend a result of Etnyre [1999] 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.