I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information.

On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in Dynamical System - Lorenz Equations (http://www.math.columbia.edu/~jb/bw-KPO-I.pdf), Theorem 6.4 states that there are Torus links which are not Lorenz links. However, in a more recent preprint on Lorenz Knots (available here: https://arxiv.org/pdf/1201.0214.pdf), Joan Birman says on page 12 of the document that all Torus links are in fact Lorenz links. Indeed, this seems to draw upon something she proved with Ilya Kofman, where they introduced a new class of links known as T-links (which appears to be a generalised form of Torus links) and proved that there was a 1-to-1 correspondence between T-links and Lorenz links.

This seems to be an obvious contradiction - am I misunderstanding something?


1 Answer 1


The point is that the two papers use slightly different definitions of "Lorenz Links".

The newer paper defines Lorenz links as links on the Lorenz template. With this definition all torus links are Lorenz links.

The older paper excluded links with a parallel cable around some component from the definition. So for example (n,n)-torus links were no Lorenz links with respect to that definition.

This issue is discussed in the introduction of the Birman-Kofman paper you mentioned.

  • $\begingroup$ Thanks, that helped! I was being silly/careless and somehow missed that comment. $\endgroup$
    – asldjk
    Aug 29, 2017 at 14:56

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