by William W. Menasco

I first meet Joan Birman in the Fall of 1985. I had been at the
University at Buffalo as an assistant professor for one year. Its
mathematics department had a laudable history of research in classical
point set topology and foundations. But, my hire showed the
department’s interest in expanding into geometric topology. With this
in mind and knowing of Joan’s work in low dimensional topology and
braids, I invited her up to Buffalo to give the department’s
colloquium. Although it was a short visit we managed to talk a good
deal of mathematics and Joan gave me a peek into the large world of
braids. She told me of the Jones’ conjecture — I would end up
pursuing it for the next

After our initial meeting, we were in steady communication for a number of months. This was before the advent of email, so we talked on the phone a good deal and used snail-mail. Additionally, we started racking up the frequent flier miles, me going down to New Rochelle to stay at the Birman household, and Joan coming up to Buffalo and staying at my home. Our visits were filled with math, cooking and conversation over family meals. I became good friends with Joe, Joan’s husband, and Joan saw my wife, Melissa, and me raise our two boys, Timothy and Ryan.
Early in our discussions I asked Joan if she had a big “dream”
conjecture. Her reply was that she hoped to be able to use closed
braids in solving the knot classification problem — a simple version
of this is, give an effective algorithm for determining when two knots
in

This idea of canned isotopies seemed reasonable to me. My doctoral thesis had been on incompressible surfaces in the complement of alternating knots and I had Tait’s flyping conjecture on the brain for some time. Working out some low index examples we quickly realized that the calculus for 3-braids had to be just braid preserving flypes. But, the problem with all mathematics is how to turn a had into a proof. It took us some time to develop traction. By the summer of 1986, we thought we had some results that we were going to present at a UC Santa Cruz braid conference, but we found a fatal flaw in our argument at the last minute and had to cancel.
Finally, in 1987 we started to make some substantial headway. Using
basic general position arguments we considered a spanning surface of a
closed braid and its intersection with disc fibers of the braid
foliation. Positioning the spanning surface so that it is transverse
to the braid axis and all but finitely many disc fibers, one obtains a
singular foliation on the spanning surface that looks like the union
of four-sided tiles, each tile containing a saddle-point where the
surface and a disc fiber intersect nontransversely. The vertices (or
corners) of these tiles were where the surface was punctured by the
braid axis. There were natural parity assignments to each tile and
each vertex. With all of this combinatorial information the question
was how to see when a braid admitted a possible canned isotopy. We
realized that the singular foliation had natural local manipulations
that corresponded to braid isotopies, stabilizations,
destabilizations, and exchange moves. We also realized that we had
stumbled onto technology that was essentially what
Daniel Bennequin had developed in his seminal work on braids and contact geometry
[e1],
the one difference being his singular foliation was
induced by the standard contact plane field in
One key observation that Joan made was the tiling nature of the
singular foliation yielded an Euler characteristic equation which
determined when an exchange move or destabilization must occur. With
the singular foliation and this Euler characteristic equation in hand
we cranked out a “Studying links via closed braids” series of papers
[5],
[3],
[4],
[7]. Specifically, we established that if
at a fix braid index a link type has infinitely many distinct closed
braid representative isotopy classes then all but finitely many of
them were related by exchange moves. For closed braid representative
of the
Since then several scholars have modeled their investigations on this strategy of having a surface tiling structure powered by a Birman-M Euler characteristic equation. Both Peter Cromwell [e2] and Ivan Dynnikov [e8] used this strategy in their works on arc presentations of knots. Tetsuya Ito employed the tiling/Euler-equation combo his very beautiful work which made connections between the braid group orderability and essential surfaces in the knot complement [e11], [e10], [e14]. And, I will mention others shortly.

In our paper on 3-braids [8] we finally came back to establishing the closed braid calculus for oriented links represented by closed 3-braids. We were interested in proving what we realized from the start of our work, that calculus for 3-braids had to be just braid preserving flypes. In fact, there are two types of braid preserving flypes — positive flypes and negative flypes. This fact would become important later when we started to think about applications of our work to knot theory in the contact geometry setting. What allowed us to classify closed 3-braids was an observation that was made originally by D. Bennequin — a minimal genus spanning surface of 3-braid intersects any 3-braid axis exactly three times [e8], [3]. Our argument then focused on an analysis of how two axes representing distinct 3-braid conjugacy classes could possibly intersect the same minimal genus spanning surface. The upshot is that every closed 3-braid link has either one unique conjugacy class, or two. And, if two then the two classes are related by a braid preserving flype. (See the figure above.) Later Ki-Hyoung Ko and Sang-Jin Lee determined which 3-braids admitted both positive and negative flypes [e4]. This became important for our contact geometry applications, which I will come to momentarily.
The classification of closed 3-braids partially opened the
stabilization black box. The 3-braid preserving flype corresponded
to a fixed stabilization sequence — stabilize positively (resp.
negatively), followed by a particular 4-braid isotopy, followed by a
positive (resp. negative) destabilization for a positive (resp.
negative) flype. (See [10] for sequence.) So the closed
3-braid calculus was finite — one canned isotopy, or one
template. By 2002 we thought we were ready to write down the
details of the Markov Theorem without Stabilization (MTWS),
which states the following. Let
The most immediate application of the MTWS was to contact geometry and
knot theory. A contact structure on a smooth 3-manifold is a smooth
2-plane field in the tangent bundle that is completely
nonintegrability. The standard contact structure for
A few additional remarks are of interest. First, in [10] Joan and Nancy Wrinkle showed that positive stabilizations, exchange moves and positive braid preserving flypes were transverse isotopies. (Negative stabilizations alter the self linking number so they cannot be transverse isotopies.) Second, since there was a large community of contact geometers working with Floer homology in an effort to produce transverse and Legendrian knot invariants, Joan thought it would be a good idea to produce a table of low crossing 3-braids which were not transversally simply [14]. Recent work of Lenhard Ng, Peter Ozsváth and Dylan Thurston [e9] has established that certain closed braids admitting a negative flype can be distinguished by such Floer homology invariants, but the 3-braids of this table are still resistant to such invariants.
In 2012 the validity of the Jones’ conjecture was finally established, first by Dynnikov and Prasolov [e12] using further innovations of the singular foliation technology in the arc presentation setting, and then by Doug Lafountain and myself [e13] using further innovations of clasp annuli setting. Both approaches use an Euler characteristic formula coming from a singular foliation tiling.
I have heard it said that a successful career in mathematics is at least two good ideas. From her initial landmark book “Braids, links, and mapping class groups”, to her work on the Jones’ polynomial, to her work on the Vassiliev’s invariants, to her work with me on singular foliations, and more, I would say that a distinguished career comes from having a eye for where the mathematics is and where it should go.