M Theory Lesson 66

Recall that Joan Birman et al studied knots in the Lorenz template with two generating holes X and Y. So knots are expressed as words in X and Y. In Robert Ghrist's paper Branched two-manifolds supporting all links he shows that the template $\mathcal{V}_0$ on more letters contains an isotopic copy of every (tame) knot and link. More specifically, for a parameter range $\beta \in [6.5,10.5]$ every link appears as a periodic solution to the equation which is used to model an electric circuit. This is cool stuff. In M Theory we like ribbon diagrams which are twisted into loops like in the Lorenz template diagram. The universal template $\mathcal{V}_0$ can be embedded in an infinite sequence of more complicated templates, which in turn are embeddable in $\mathcal{V}_0$. Ghrist also considers flows arising from fibrations, such as the 1-punctured torus fibration for the figure 8 knot complement. This fibration flow is also an example of a universal flow.

I was quite intrigued when a mathematical biologist at a conference told me recently that no one really knew why DNA had four bases rather than two. Apparently it isn't clear why self-replicating molecules fail to adopt a binary code in X and Y. Somebody else muttered something about hydrogen bonds and then, inspired and ignorant, I started rambling on about knot generation in templates. After all, DNA molecules need to know how to knot themselves.