#### 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 __\( 20+ \)__ years. Simply stated, the conjecture
claims that the algebraic length of a minimal index closed braid
representative of an oriented knot is an invariant of the knot.
Dropping her off at the airport, Joan said “maybe we can do some work
together”. Famous last words. We went on to co-author 13 papers and
two more manuscripts where we were joined by additional collaborators.

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 __\( S^3 \)__ are isotopic. Garside’s solution to the conjugacy problem
supplied the means for deciding when two __\( n \)__-braids were in the same
conjugacy class; that is, when one given __\( n \)__-braid is isotopic in the
complement of its braid axis to a another given __\( n \)__-braid. Markov’s
theorem told us that two closed braid representatives of the same knot
type are isotopic through a sequence of stabilizations,
destabilizations and braid isotopies. Markov’s theorem had been
essential to Vaughn Jones’ proof that his knot polynomial was a knot
invariant. But, stabilization-destabilization sequences were a “black
box”. What did stabilization buy you? In one of Joan’s earliest
letters to me she wrote: “In my book, page-100, example 2-27, you’ll
find a sequence of moves which realize an exchange by going up [in
index] once and then down [in index] again.” [1] This one
example, which I think was originally due to
Jim Van Buskirk, was the
basis for Joan’s hunch that there existed a * closed braid
calculus* — a set of “canned isotopies” that allowed one to move
between closed braid conjugacy class representatives of the same knot
type without having to stabilize. Joan’s page-100 * exchange move*
was just one such canned isotopy. (See the next figure. See [4] for
a full discussion of the calculus.) Joan’s focus on this one example
illustrates a quality that I have always recognized and appreciated.
She has a eye for where the mathematics is and where it should go.

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 __\( \mathbb{R}^3 \)__.

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 __\( k \)__-component unlink we established that through a sequence of
braid isotopies, destabilizations and exchange moves, one could obtain
the __\( k \)__-braid unlink. In [2] we replaced the spanning surface
with an __\( S^2 \)__ illustrating that the link was a split or composite
link. Specializing the singular foliation and Euler equation to this
surface we again established that all that was needed for moving from
an arbitrary closed braid representative to one where the links
decomposing component were readily visible were braid isotopies and
exchange moves. This pattern of exchange moves being the primary
simplifying canned isotopy held where we replace __\( S^2 \)__ with an
essential torus in the link complement [9].

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 __\( (m,n) \)__ be any pair of positive
integers with __\( 0 < n < m \)__. Let __\( \beta_1 \)__ and __\( \beta_2 \)__ be closed braid
representative of the same link type with the braid index of __\( \beta_1 \)__
being __\( m \)__ and the braid index of __\( \beta_2 \)__ being __\( n \)__ which is minimal
for the link type. Then there exists a finite collection of isotopy
templates depending only on __\( (m,n) \)__ which can be used in an isotopy
sequence of closed braid that move __\( \beta_1 \)__ to __\( \beta_2 \)__. And, here
is the kicker, the braid index of this sequence is nonincreasing
[6]. The proof of the MTWS was a detailed analysis of a very
simple proof of the Markov Theorem that we put forward in a short
paper [11]. The salient new technology that we introduced in
this paper was the * clasp annulus* — annuli that correspond to an
isotopy between boundaries but which also has clasp
self-intersections.

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 __\( {\mathbb R}^3 \)__
corresponds to the kernel of the 1-form, __\( \alpha = d z + r^2 d
\theta \)__, in cylindrical coordinates and the standard contact structure
for __\( S^3 \)__ can be seen as the plane field that is normal to the Hopf
fibration of __\( S^3 \)__ by __\( S^1 \)__’s. We incorporate contact
structures into knot theory by considering knots that are either
totally transverse to the contact plane field — * transverse
links* — or totally tangent to the plane field — * Legendrian
links* — and isotopies that preserve these qualities. Daniel
Bennequin in his seminal work
[e1]
showed that every transverse
link was transversely isotopic to a braid representative. An immediate
invariant of a transversal knot class is the * self linking
number*. It was immediately clear to Joan and me that the isotopy
template coming from the MTWS could either correspond to a transversal
isotopy or not. A necessary condition of a template to be transverse
was that it must preserve the self linking number. For example,
Yakov Eliashberg in 1998 had established that the transversal classes of the
unknot were classified by the self linking number. He had speculated
in conversation to Joan that knot type plus self linking number was
all that was needed to classify transverse isotopy classes of knots.
Then Joan showed him 3-braids that admitted a negative flype, and
Eliashberg agreed with us that any 3-braid that admits only a
negative flype * had* to have (two) distinct transverse
isotopy classes with the same self linking number; that is, such
3-braids were not * transversally simple*. After establishing the
MTWS, we were able to adapt our technology to make this * had*
into a proof [13].

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.