#### by Joan Birman

My collaboration with Bob Williams began in January 1982 and continued for about 3–4 years. I refer to him as Bob, because our relationship was informal from day 1. It was both interesting and highly rewarding for both of us, and while we did not work together after that, we remained good friends, with a friendship that had its roots in the pleasure that we both felt when we made a new discovery in mathematics.

I had gone to the 1982 winter meeting of the AMS for a committee meeting, but there was free time too, so I wandered over to listen to one of the invited talks. Arriving early, I sat down next to Bob Williams. We introduced ourselves and began to chat. He told me that he worked in dynamics, more particularly on chaotic flows in differentiable dynamical systems, an area about which I knew very little. I told him that I was interested in braids, knots and links in three-space. That prompted him to dig into his pocket and remove a piece of paper with a picture of what I learned later was a Lorenz knot. It was a simple closed curve in three-space, given by a projection onto a plane, with the six double points. It was easy to see that there were two loops that could be pulled tight to remove two of the double points (albeit at the expense of losing the interesting pattern that was evident in the original picture), but at first glance an eraser was needed to see whether all the double points could be removed. He asked “Is it knotted?” I answered, almost immediately, “Yes.” He asked me how I knew? I explained, and our collaboration was on its way!

How did I know? The representation of knots and links by closed braids had played a central role in my own research, and I could see that the natural generalization of the knot or link in the picture in Bob’s pocket was in the form of a (slightly concealed) closed braid with all crossings having the same sign. By a stroke of luck, I happened to know that the crossing number minus the number of braid strands (3 in his example) plus 1, was a knot type invariant (essentially the genus). In the example he showed me the genus was 1, not 0 as it would have been if it was the unknot, and that was how I knew it was not the unknot.

His example interested me immediately, because I had been searching
for interesting families of knots and links that were given by
projections, but not necessarily by minimum crossing projections, and
it was immediately clear that Bob was showing me such a family. A word
is in order about the classical knot tables, reproduced at the end of
Dale Rolfsen’s
excellent textbook
[e2].
The tables list
knots by choosing a minimum crossing number planar projection,
beginning with crossing number __\( 3,4,\dots \)__ and going up to 8
crossings. At the time Bob and I met I knew that three of our
colleagues
[e4]
were using modern computers in an attempt to
extend the tables in
[e2]
to __\( 9,10,\dots \)__ crossings, only to
discover that by the time they reached crossing number 17 there were
1,701,936 distinct knot types, each defined by a picture! Very little
structure was discovered. I had became convinced, at the time when I
met Bob, that minimum crossing number was probably not a good measure
of the complexity of knot type, and Lorenz knots were a beautiful
example to show that my instincts were on the money.

I learned, from Bob, that the natural generalization of the knot in
the picture that he had shown me
is the family of closed orbits in the
solutions to a very simple set of
three ODEs, with space variables
__\( x,y,z \)__ and time variable __\( t \)__. The system is nonlinear, but it’s very
close to a system of linear equations in __\( x,y,z,t \)__. Bob explained to
me that the solutions are a family of simple closed curves in __\( R^3 \)__
that may be projected, simultaneously and disjointly, onto a “branched
2-manifold”, now known as a *template* (we called it a
*knot-holder*), and even more that the template determined an infinite
family of knot and link diagrams that we called *Lorenz links*.

A short time later Bob told me about a second family of knots that he
had begun to study, that were * also* determined by a template in
__\( R^3 \)__. The
figure 8 knot __\( F_8 \)__ embeds in __\( R^3 \)__, and since __\( R^3 \)__
embeds in __\( S^3 \)__, the complement of __\( F_8 \)__ can be regarded as a subset
of __\( R^3 \subset S^3 \)__. He knew that __\( F_8 = \partial W^2 \)__, where __\( W^2 \)__
is a 2-manifold with boundary that is embedded in __\( R^3 \)__, and that
__\( S^3\setminus F_8 \)__ has the structure of a surface bundle with __\( W^2 \)__ as
fiber. It follows that if one can find (as Bob had started to do) a
surface that it bounds in __\( R^3 \)__, then one should be able to follow
points on that surface as one pushes the fiber off itself and around
in a circle and then back onto itself, and trace out their orbits,
which would be knots and links in __\( R^3 \)__. When I visited him in his
office one day I was astonished to see that entire project under way,
a massive construction made up out of strips of paper and string, held
together with paper clips and glue, and suspended from his office
ceiling! So it turned out that my new friend had been thinking about
knots and links very seriously for quite some time before we met! See
[1],
for example, where he worked out in detail the
symbolic dynamics that allowed one to determine the closed orbits in
the Lorenz attractor, pointing out that most of the orbits appeared to
be knotted.

A short time after that initial meeting, Bob told me about the templates and associated branched surfaces and the tools of symbolic dynamics that were at the heart of his work. In turn, I told him about the tools that I knew from knot and link theory. Working together, we proved (among other things) some interesting properties of Lorenz knots and links:

- There are infinitely many inequivalent Lorenz knots. These include Lorenz knots of arbitrarily high genus, although for fixed genus
__\( g \)__only finitely many distinct knot types occur. - Every Lorenz knot and link is fibered.
- Every algebraic knot is a Lorenz knot, and some algebraic links are Lorenz links. In particular, all torus knots occur but some torus links do not.
- There are Lorenz knots that are not iterated torus knots; there are iterated torus knots that are Lorenz but not algebraic.
- Every Lorenz link is a closed positive braid, however there are closed positive braids that are not Lorenz.
- Every nontrivial Lorenz link of two components is unsplittable, also the algebraic and geometric linking numbers are positive and equal.
- Nontrivial Lorenz knots and links are nonamphichiral.
- Nontrivial Lorenz links have positive signature.

Most important, to me, was the fact that we had discovered a family of
knots and links that had immediate structure, and they were defined by
a family of link diagrams that
*did not* have minimum
crossing number. As for Bob, he was the first mathematician in his
area to systematically study the knot and link types of the closed
orbits in a differentiable dynamical system, and to show that the knot
and link types of the closed orbits in a dynamical system had
independent interest as a new family of knots and links.

Following our success with Lorenz links, we began to study the
periodic orbits in the complement of __\( F_8 \)__, first completing Bob’s
project of finding its template, and investigating the family
__\( \mathcal F_8 \subset (R^3\setminus F_8) \)__ of
figure 8 knots and links (see
[3]).
To our
astonishment we were unable to find any results like the ones we
quoted above for Lorenz knots and links! We didn’t know what to make
of it, until
Rob Ghrist
[e3]
pointed the way a few years
later: Ghrist proved that there are two types of templates associated
to flows in __\( R^3 \)__: In the generic case (e.g., the flow associated to
__\( \mathcal F_8 \)__) the closed orbits are ALL knots and links. But in very
special cases (e.g., the Lorenz flow) the knots and links are a unique
family, and they characterize the flow, which is a typical example of
a flow that had been dubbed “chaotic”, but in fact has a great deal
of structure. Moreover the knots and links characterize the flow. Since
Lorenz links had their origins in the ODEs that were used in 1962 to
predict the weather
[e1],
that was very interesting.