#### by Rob Kirby

##### Editor’s Note

*Birman’s early work with
Hugh Hilden
is not discussed here because it is already covered in two other articles in this volume:
Birman’s essay on
her favorite paper
and Dan Margalit and Rebecca Winarsky’s article
“The Birman–Hilden theory”.
*

#### Magnus and braid groups

**Rob:** *Can you tell me how you became interested in braids?*

**Joan:** Braids were a natural interest for me. My grandmother was
very thrifty, and would take apart outgrown sweaters, soak the wool and
hang it over the faucet of the bathtub to dry. I would help her wind the
recovered wool into a ball, and she would reuse it to make a small sweater
or scarf or hat. I remember holding my hands out very wide, to hold the
partially wound skein of wool, so that she could wind it into a ball.
So I learned about braids and knots from daily life when I was very young,
and had thought about how to “untangle” braids and knots.

**Rob:** *But how did you learn about braids in mathematics?*

**Joan:** From a language requirement for admission to research at
NYU’s Courant Institute! My exam was to translate
Artin’s
1925 paper on
braid groups
[e1].
The paper was a fine choice. I was very taken with the idea that group
theory could capture all that visually appealing topology, and give it new
and unexpected mathematical structure. So in the course of locating the
verbs in those very long sentences, I became hooked by the mathematics.

**Rob:** *That’s a good explanation.
Still, having worked in applied mathematics
out in the real world, it’s interesting that when you went to
NYU, you did not choose an applied topic.*

**Joan:** You’re right. I had been working in industry before
our three children were born, but for some years had been mainly at home
taking care of them. My plan when I began graduate studies part-time, at
night, was to prepare for an eventual return to the job I had enjoyed.
With that in mind, NYU’s Courant Institute of Applied Mathematics was
a natural choice. It was an excellent graduate school, and it had a
truly “open admissions” policy. If you wanted to learn something that
they were teaching, you just registered, paid tuition, showed up and did
the work. Even more, my husband Joe was a tenured professor in the NYU
Physics Department, and free tuition for spouses was one of his benefits.
So there were many reasons for me to do my graduate work at Courant.
I should add that when I started, I did not intend to get a PhD, just to
bone up on math that I had forgotten and hopefully pick up some new tools.

Yes, a topic in applied mathematics would have been a natural choice. Moreover, there were very few people at Courant who were not working in various aspects of applied mathematics. In spite of that, the courses that I took, in all areas of mathematics, were excellent, so I was exposed to much more than applied mathematics.

In graduate school I had taken three or four courses on differential equations, and one in Calculus of Variations, but when I went to speak to the faculty members who had taught those courses, to select a thesis advisor, the topics they suggested just didn’t grab me. For example, when I spoke to Louis Nirenberg, who had been a favorite teacher, and who I knew was a fine mathematician, he asked me a very good question. He said, “Do you like inequalities?” And I said, “You know, I really don’t.”

**Rob:** *Whoops.* [Both laugh.]

**Joan:** I also spoke with
Jürgen Moser,
but the topic he suggested
did not excite me either. I had liked the graduate topology course that
I took very much, so I went to speak to
Kervaire
about possibly working
on my thesis with him, but he didn’t want to work with me. He said, “You
don’t know enough topology”, which was certainly true (and continues to
be true to this day).

Magnus was another Courant faculty member who did not work in analysis or PDE or applied math of various sorts. I had taken a course in group theory with him, and had been his TA at one point. He must have noticed my interests, because when I went to speak to him he suggested a topic that was instantly appealing to me.

**Rob:** *Okay, you gave a perfectly good reason for choosing
to work with Magnus: the topic he suggested appealed to you!*

**Joan:** Yes, and for me following my own interests has always been
the best way to chose a new problem to work on.

To return to my thesis, at our first meeting Magnus told me about
the mapping class group of a torus with punctures. He said that if
you removed one point the mapping class group was __\( \operatorname{SL}(2;\mathbb Z) \)__ and
that he had worked out a presentation for the mapping class group of a
twice-punctured torus. He suggested that I think about a presentation,
in the case of the torus, if you removed three or more points.
He also mentioned that he had noticed a paper by
Fadell
and
Neuwirth
[e3]
that gave a new way of thinking about braids, and suggested that I look at
that paper. I should add that I didn’t understand
[e3]
at all,
at first, and pestered one of Kervaire’s graduate students, whose name I
have completely forgotten, for help. Magnus had not been much of a help in
that regard. The paper said that Artin’s braid group was the fundamental
group of a configuration space. It took me quite some time to understand
that, but when light dawned it opened a door to a whole new world. Later,
I learned that the configuration space viewpoint was new even to Artin,
and that when he heard about it from Fadell or Neuwirth, and understood it,
his response was simple: he said ‘Well, I never thought about it that way!’.

As I began to think about the mapping class group of the punctured torus,
I realized that it was related to things I had learned in my first year
graduate topology course, and also that there was no reason to restrict to
the case of a punctured torus. The mapping class group of any surface
__\( X \)__ is __\( \pi_o(\operatorname{Diff}X) \)__, a definition that I liked because I had
learned about long exact sequences of homotopy groups, and the mapping
class group was a homotopy group. So if
__\[ X = S_{g,n} ,\]__
a compact surface
of genus __\( g \)__ with __\( n \)__ punctures, one simple version of my problem was to
identify the kernel of the homomorphism
__\[ \phi_n:\pi_o(\operatorname{Diff}S_{g,n})
\to \pi_0(\operatorname{Diff}S_{g,0}) \]__
defined by filling in the __\( n \)__ points. I was
able to construct a long exact sequence of homotopy groups that contained
the homomorphism __\( \phi_n \)__ and the three groups
__\[ \pi_o(\operatorname{Diff}S_{g,n}), \quad
\pi_0(\operatorname{Diff}S_{g,0}) \quad\text{and}\quad
\operatorname{kernel}(\phi_n) .\]__
In my thesis, I proved that the kernel of __\( \phi_n \)__ is isomorphic to the
__\( n \)__-strand braid group of the surface mod its center. So braids, newly
generalized, had appeared in my life again: first, via configuration spaces,
and second as braid groups of surfaces, a concept that had been natural,
once one understood the aforementioned paper of Fadell and Neuwirth.

In the case I considered the kernel is __\( \pi_1(S_{g,0},\star) \)__ mod its
center, and a new problem presented itself: to “see” the kernel as a
concrete subgroup of the mapping class group. In the simplest case, __\( n=1 \)__,
the problem was to construct an element of __\( \operatorname{Diff}(S_{g,0}) \)__ that drags a
point on the closed surface __\( S_g \)__, say the base point __\( \star \)__, around a
closed loop that realizes an arbitrary element of __\( \pi_1(S_{g,0},\star) \)__.
The memory of the joy I felt, and even where I was standing when the
“aha” moment came and I realized how to construct the map, was a very
powerful experience. That map, which is known as the *point-pushing
map*, has been studied in detail quite recently, e.g., see
[e15].

My thesis was about the simplest example of what is now called the Birman exact sequence. That sequence has been generalized in many ways.

**Rob:**
*You have your name attached to quite a few things.*
*How did you happen to write a book about braids and
mapping class groups?*

**Joan:** My research after graduate school began with joint work with
Mike Hilden
on mapping class groups of *closed, i.e., no longer punctured*
surfaces
[1].
He was a graduate student at Stevens Institute,
where I had my first job after graduate school. We were both unknown in the
larger math community, and so perfectly free to explore whatever interested
us. When our joint work became known, it lead me to an opportunity to teach
a graduate course at Princeton, where I was free to select the topic, and my
choice was to put together all the things I knew relating to braids, links
and mapping class groups. Then the possibility arose of writing a research
monograph that was based on the course, and I immediately said “yes”.

As it turned out, Rob, the mathematics around braids had been relatively
undeveloped for a very long time when I began to work on braids. Those were
the days of the Bourbaki, when the idea of a concrete visual interpretation
of any part of mathematics was regarded as being, by its very nature, in
need of work to make it appropriately abstract. You were at the height of
your own career when I got started, so you surely know that, at the time,
braids were regarded as a backwater of topology, yet they had already
appeared in many ways in mathematics. Later I realized they also were
present in the knotted orbits of flows on __\( S^3 \)__, and must even have been
implicit in operator algebras, encoding the way that one type __\( \text{II}_1 \)__
factor is sits inside another type __\( \text{II}_1 \)__ factor.

**Rob:** *Well, it’s a real art, to discover a key to an unknown
garden with lots of low-hanging fruit.*

**Joan:**
I was very lucky. I chose to study braids because they appealed to me.
It was a time when very few topologists were interested in them, so there was
time to explore the mathematics, free of the pressure of lots of competition.
And then, it developed that braiding is a very fundamental phenomenon
in mathematics and, indeed, in nature.

#### Dennis Johnson, the __\( \mu \)__-invariant, and ~~Bob Craggs~~

**Rob:** *How did you come to talk to*
*Dennis*
*in the first place?*

**Joan:** I don’t remember when and how I met Dennis, but I think
it was after I joined the Barnard-Columbia faculty in 1974. Dennis
and I had common interests in mathematics, we began to talk and
quickly became friends. He knew of my interest in the Torelli
subgroup of the mapping class group, and the then-open question of
whether it was finitely generated, and I provided an audience for him
when he began to study the Torelli group. You can see from the
letters that he wrote to me1
how eager he was to talk about his work.
He called me many times on the telephone on days when he knew that I would be
at home working, as he was making his discoveries about the Torelli group.
Somewhere during that period we also began to work together on what we
called *symplectic Heegaard splittings*, but later he insisted that
that work be “for the drawer”, and while I didn’t really want it to be
that way I felt there was no option but to agree.

**Rob:** *So how did the
Birman–Craggs homomorphisms*
[1]
*come about?*

**Joan:** The classification of 3-manifolds was a very important
problem in low dimensional topology at that time. I knew you could construct
every 3-manifold by Heegaard splittings, and some of them by surface bundles,
so I thought it might be possible to understand 3-manifolds by studying
mapping class groups of surfaces. The idea that you could use actions on
2-manifolds to learn about 3-manifolds seemed beautiful and intriguing to me.

I don’t remember how I met
Craggs,
but it was probably at a conference.
My dream had been to use knowledge about diffeomorphisms of surfaces to learn
new things about 3-manifolds, but he had been thinking about the same matter
in a different way: that known structure regarding 3-manifolds ought to be
reflected in actions on surfaces. So our interests coincided. Ultimately,
we found a way to put together our two viewpoints, using them to construct
a family of homomorphisms from the Torelli group onto __\( \mathbb{Z}/2\mathbb
Z \)__. The initial construction depended on the choice of a pair of elements
in Torelli, giving us two Heegaard gluing maps that we used to construct two
homology __\( \mathbb S^3 \)__’s that were distinguished by their Rohlin invariants.
In fact, there wasn’t a *unique* such homomorphism, as it turned out we
discovered there was a whole family of homomorphisms from the Torelli
subgroup of the mapping class group to __\( \mathbb{Z}/2\mathbb Z \)__, depending
on how you selected the original homology __\( \mathbb S^3 \)__. Craggs was
certainly the expert on the Rohlin invariant, and I was very lucky to have
him as a partner in work that related so beautifully to one of my math dreams.

**Rob:** *Okay.*

**Joan:** I knew the question of whether the Torelli group was
finitely generated was a major open problem, and was very excited by an
unexpected turn in events. I thought, “Oh, wow. maybe there are *infinitely many homomorphisms* onto __\( \mathbb{Z}/2\mathbb Z \)__, which would
imply that the Torelli group is infinitely generated!” That was the
main question in my mind, as my work with Craggs developed. But then,
one day I did some calculations relating to our work and realized there
was disappointing news: there were finitely many homomorphisms. We had
learned that the underlying idea of relating the mapping class group of a
surface to 3-manifold topology, via Heegaard splittings, was sound and new,
but it didn’t give us the result I had hoped most to get.

Dennis read our paper, and was very excited by it. Starting his work where we had ended ours, he began by counting how many distinct homomorphisms there were! His paper [e5], published in 1983, began right there, and developed into the discovery of structure in the Torelli group that is still being investigated as we speak, in 2018. At the same time, he proved that Torelli was finitely generated, and not by the expected maps (Dehn twists on bounding simple closed curves or BSCC), but by maps that were known as BP or “bounding pairs” maps. In fact, BP maps had made their appearance earlier, when I did an enormous calculation that yielded a proof that, up to conjugacy, the Torelli group was generated by Dehn twists about BSCC’s and “something else”, which I could only describe as a very long product of Dehn twists about a sequence of curves on the surface. Jerry Powell, who was working on his PhD thesis under my supervision at the time, interpreted the “something else” as a BP map, so his contribution also played a big role in Dennis’ work, which was beautiful, imaginative, and highly creative.

#### Lorenz

**Rob:** *Can you tell us about*
*Lorenz*
*attractors and your
collaboration with*
*Bob Williams**?*

**Joan:** Yes, it’s easy to pinpoint that one. I attended a winter
meeting of the American Mathematical Society (AMS), probably in 1981. I went to one of the main talks, and
while waiting for it to begin began chatting with the man sitting next to me,
Bob Williams. He said, “Oh, you work in knot theory. Let me show you this
example’. He pulled a picture out of his folder of what he later explained
was a picture of a closed orbit in the Lorenz flow on __\( \mathbb S^3 \)__.
The picture showed a knot with loads and loads of crossings but you could
see right away how to lift up pieces of it, untwist and then tighten them,
so the question was, did that process end? Knowing the answer, he asked me,
“Is it knotted?” Well, I looked at it, and I could see right away that it
was a closed braid, and that all the crossings had the same sign. So it was
a “positive braid” and I happened to have a formula in my head so that if
you knew the number of braid strands and the crossing number of a positive
braid that closed to a knot, you could compute the genus of the knot.
So a minute later I said “yes”, and that it had genus one. So that’s
how we started to talk about knotted orbits and dynamical systems. It was
Bob Williams’ idea that there should be knots. He showed me some of the
knots that occurred. I contributed the tools that allowed us to study them
as a class.

**Rob:** *So you came up with the notion of a knot holder?*

**Joan:** No, that concept was known to Bob (and had even been
described intuitively by Lorenz). I learned about it from Bob. By the
way, knot-holders have since been renamed templates, a more elegant name.
The template was a branched 2-manifold that was embedded in 3-space.
As Bob explained things to me, there was (loosely speaking) a foliation
of 3-space, and the union of *all the closed orbits* in the Lorenz
flow could be pushed, simultaneously and independently, along the leaves
of the “foliation” and onto the template. That was clearly a deep
and wonderful theorem, and I learned about it from Bob Williams, who was
a fine teacher! He wrote the section of our paper that described the
template and its properties. My contribution to that part of our paper was
to be the Chief Nag, pressing him to write down a proof of its existence
that would be acceptable in the topology community. In that sense our
paper became the go-to reference for templates. Another main new result
in our paper was the proof that the knots and links that were determined
by the Lorenz template were a new class of very special knots and links,
characterized by their properties. Among those properties are that Lorenz
links are prime, fibered, have positive signature, include all torus knots
and some (but not all) torus links.

The template had actually been described intuitively by Lorenz, in his
original and now classical paper on the Lorenz equations
[e4],
which are a very simple set of three almost linear ODE’s in three space variables
__\( x,y,z \)__ and time __\( t \)__. Lorenz was a meteorologist, initially, was aware
that certain families of ODE’s acting on 3-space had very complicated
closed orbits. His contribution
[e4]
was to reduce one of those
families of ODE’s to a related family of very simple ODE’s that could be
studied with the help of computers. Lorenz was a highly creative scholar
with very deep insight into structure.

**Rob:** *Right.*

**Joan:** So all those knots and links are disjointly embedded in the
Lorenz template, which is a branched 2-manifold that embeds in __\( \mathbb
R^3 \)__, also (with a few exceptions) every closed orbit on the template is
a closed orbit in the Lorenz flow on 3-space. We wrote one paper that was
quite successful because once we knew that the knots on this template were
all positive, we were able to compute, from properties of the template,
the minimum number of braid strands for the knot type, i.e., the *braid index*. So we got a formula for the braid index, and that was quite
wonderful. Lorenz knots turned out to be a family of very high braid index
knots, whereas the knots in the tables of knot diagrams all have very low
braid index, so in that sense they were very new. The fact that we could
actually compute their minimum braid index was wonderful. We learned
a lot about Lorenz knots, and in fact did so well that we then said,
‘Great, let’s look at another example’.

The new example that Bob proposed for study was a family of closed
orbits in the complement of the figure-eight knot in __\( \mathbb R^3 \)__.
That knot is fibered, and its monodromy is pseudo-Anosov, indeed
Thurston
introduced it to the topology community as the famous
__\( \bigl[\begin{smallmatrix}2 &1\\1 &1\end{smallmatrix}\bigr] \)__ map.
A theorem of Thurston told us there were many periodic points in the
monodromy map of any pseudo-Anosov map acting on a surface, which is this
case was a torus __\( \mathbb T^2 \)__ with one disc removed. The fibration was
the suspension of that monodromy map, and it determined a flow on
__\( \mathbb R^3 \setminus \mathcal K \)__, where __\( \mathcal K \)__ was the figure-eight knot,
embedded in 3-space. Our idea was to start with points on the fiber, we
would follow them around as the surface was moved through the fibration and
back onto the initial fiber. Williams had constructed a huge model of the
fiber, in his office, and wanted to do this concretely, he really wanted to
“see the flow”. I agreed with him that there were plenty of closed
orbits in that flow, and that they were embedded in 3-space, and that in
principal there ought to be knots. So once again Williams had the tool (if
you want to be a very charitable and call the monster he had constructed
out of string, paper and paper clips a tool) to describe the template,
and I had tools from knot theory, and we hoped to study the closed orbits.
But, to our great surprise, while we could describe the template, just as
we had described the template in the Lorenz flow, we didn’t find a single
theorem that would distinguish any one knot carried by the template from
any other, or from arbitrary knots in 3-space. We were very surprised. Here
was one case where there was a huge amount of structure and another case,
that seemed to be very similar, in which we found no structure at all.

As it turned out, the difference between the two cases was fundamental and
the explanation was elegant and beautiful.
It was discovered by
Rob Ghrist, who wrote the key paper near the beginning
of his own highly successful career. He proved that *every* knot and
link appeared as a periodic orbit in the flow determined by any fibered knot
in __\( \mathbb R^3 \)__ with pseudo-Anosov monodromy
[e10]
[e12].
So the reason we couldn’t find any special properties to distinguish the knots
in the flow we had studied was because there were no special properties!
*All knots and links* were right there, embedded simultaneously
and disjointly on the template Bob had constructed with scissors, paper
and paper clips! The figure-eight knot is “universal” in that sense.
The Lorenz flow was quite different. The knots in the Lorenz flow, and
others like it, are determined by closed orbits in the solutions to nonlinear
differential equations, and those knots contain information about the flow.

**Rob:** *Interesting.*

**Joan:** Fast forward to 2006, when
Étienne Ghys
gave a plenary
talk at International Congress of Mathematicians (ICM) 2006
[e13].
Perhaps you were there, and heard
him speak. I was not, but several people told me about it, because he
discussed Lorenz knots and my by-then-old work with Williams, and showed
some beautiful slides. The principal topic of his talk was the so-called
*modular flow*, that arises in number theory, and his proof that the
closed orbits in the modular flow, which is a flow on the complement of a
trefoil knot __\( \mathbf T \)__ in __\( \mathbb S^3 \)__, are in 1-1 correspondence with
the closed orbits in the Lorenz flow. I don’t want to take the time to say
more about this, except to note that Ghys’ trefoil knot __\( \mathbf T \)__ was not
the trefoil that Bob Williams had shown me in 1981 the day we first talked
at that AMS meeting, in fact it is not an orbit in the flow, although it is
a simple closed curve in __\( \mathbb S^3 \)__. So where is __\( \mathbf T \)__ to be
found, and what makes it different from the trefoil that Bob Williams had
shown me at that AMS meeting in 1981? *That matter* is explained in a
2017 paper by
Tali Pinsky
[e16].
The knot __\( {\mathbf T} \)__ is the
union of singular orbits in the solutions to Lorenz’ ODE’s in __\( x,y,z,t \)__.
There are three clear singular points, and experimental work suggests that
each singular point is connected to the next and then to the third one,
through singular leaves whose union is Ghys’ trefoil.

**Rob:** *I see. That’s quite a story!* [laughs]

**Joan:** Yes, and it’s far from over, because Tali’s work is both
partially descriptive, and suggestive that much more is there to be done
than I could possibly mention today.

#### The Jones polynomial

**Rob:** *Joan, can you tell us about the origins of your
connection to*
*Vaughan Jones*
*and his knot polynomial?*

**Joan:** Sure, I’m happy to do that. In the spring of 1984 I
had been working with
Caroline Series,
who was on sabbatical and had
spent some time at the Institute for Advanced Study (IAS) in Princeton,
New Jersey. She met Vaughan Jones there, He told her about his Hecke Algebra
representations of Artin’s braid group that he had discovered. He knew they
gave representations of Artin’s braid group, and he had read Artin’s paper,
and he was searching for someone who could help him to understand their
meaning better. Caroline said, “Well you must go and talk to Joan about
this”, and that’s why he contacted me.

**Rob:** *Did he know about your book?*

**Joan** Yes, because he included it as a reference in a paper he
had presented
[e6]
at a conference in Kyoto Japan in July
1983. He proves, in that paper, that his Hecke algebra representation of
the braid groups __\( B_n, n\geq 2 \)__ was reducible, and included as a summand
the Burau representation of __\( B_n \)__. He had discovered a trace function on
the Hecke algebra. He knew (in a vague way, I think) that the Alexander
polynomial of a knot or link was determined by the Burau representation of
a braid that determined it.

Following Caroline’s suggestion, he contacted me and we arranged to get together in my office at Columbia on Monday, May 14, 1984. At the end of that meeting we arranged a second meeting on Tuesday, the 22nd, just eight days later. (I know both dates precisely because I had made a note of both meetings in my 1984 “daily reminder”.) I happen to have kept all my old little books, which date back to the mid 1970’s.

At our first meeting I told Vaughan about knots and links being formed by
closed braids, and I told him about Markov’s Theorem.
He told me about his Hecke algebra representation of __\( B_n \)__. He was
very aware of the curious fact that a representation of the braid group
should have appeared in connection with type __\( \text{II}_1 \)__ factors, and eager
to understand more. I don’t know whether he knew about Markov’s Theorem.
After our first meeting, he got to work. In between those two meetings we
may have talked on the telephone once or twice, but the main point is that
by the time our second meeting began Vaughan knew he had a knot polynomial.

**Rob:** *Vaughan wrote a letter to you on May 31st, which
we include in this volume*
[11].
*The letter starts: “Dear Joan, First of all, my deepest thanks
for putting me
onto this. None of it would have begun had it not been for our seemingly
unproductive first meeting. Let me begin by summarizing what we need from
operator algebras,” and then he goes into the first theorem “For every* __\( t \)__
*greater than zero, …”*

**Joan** Yes, moreover the material at the beginning of that letter
was almost exactly what Vaughan had told me at our first meeting.
When two people in different areas of mathematics begin to talk there
is always a problem that they speak different languages, and that the
dictionaries are either poor or nonexistent, and that was the case for us.
When I asked him, at our first meeting, whether his trace function was a
matrix trace on his representations of __\( B_n \)__ his answer to me was “no”.
But I realized, much later, that in fact it
was a weighted sum of traces on each of the irreducible components. So it
was not a matrix trace, but it was a class invariant.
I did not appreciate that at our first meeting.

**Rob:** *Ah. Did you realize that first or did he realize that?*

**Joan** He answered the question that I asked. If I had asked him,
“is the trace a class invariant on __\( B_n \)__?”
he would have said “yes,” and we might have discovered the polynomial
that day, but instead, I asked “is it a matrix trace?” and he said “no.”

**Rob:** *I see. But fortunately that only slowed things down
by a week.*

**Joan** Correct. Vaughan knew that he had a class invariant, and
when we discussed the Markov Theorem he must have realized that his class
invariant behaved differently
under positive stabilization and negative stabilization, but in order to
get a link invariant the behavior had to be the same. The key new
insight that he had during the week between our first and second meetings was
that he could rescale his representations so that positive stabilization
and negative stabilization changed the trace in the same way. And lo and
behold he had a link invariant.
I should say that ideas like that rescaling might sound trivial, but I
know they are not. When the insight comes, it’s one of those “aha!”
moments that we all treasure.

When we met the second time, he began by telling me “Look, I rescaled my
representations of __\( B_n \)__, and now I have a polynomial invariant.
But it must be the Alexander polynomial.”

**Rob:** *How quickly did you realize that it was not a known
invariant, that it was not the Alexander polynomial?*

**Joan:** It took about 30 minutes, at the start of our second meeting,
to show it was not the Alexander polynomial, and two or three hours to show that
his polynomial was different from the Alexander polynomial in a deep way.
At the beginning of our second meeting, I said to him, “Well, let’s compute
it on the trefoil and on its mirror image,” that is on the closures of the
braids __\( \sigma_1^3 \)__ and __\( \sigma_1^{-3} \)__. I chose those examples because (i)
I knew that the Alexander polynomial could not distinguish them, and (ii) the
braids __\( \sigma_1^3 \)__ and __\( \sigma_1^{-3} \)__ were the simplest possible examples,
they were 2-braids of length 3, so the calculations were very simple.
Lo and behold, in a few minutes we learned that his invariant was not the
Alexander polynomial, because it took different values on the trefoil and
its mirror image.
We went on to more subtle examples. I knew about
Kinoshita and
Terasaka’s infinite family of knots with Alexander polynomial one
[e2].
Some time earlier I had computed their closed braid representatives, for
a reason that I don’t remember, and had them put in a file in my office
filing cabinet. I took them out of the file and we tried a few of the KT
knots, and his invariant was far from trivial on those. We knew (before
lunch, if I remember correctly) that he had discovered a genuinely new and
genuinely interesting polynomial.

**Rob:** *That really must have been exciting.*

**Joan:** It was very exciting. When I went home that night I thought,
“I
can’t believe that there’s another polynomial.”
Alan Solomon, a British physicist, was at our home working with Joe that
day, he had stayed at our house. I tried to tell Joe and Alan about it,
but of course there was no
way they could appreciate it.

**Rob:** *Right.*

**Joan:** When we went out to lunch that day, Vaughan had said,
“Well, I’m going to buy you a bottle of champagne, Joan.” I said,
“Well, you know, Vaughan, I
don’t want to work on this really and I don’t want a bottle of champagne, but
I do want you to give me adequate credit,” and he was very generous about
giving me credit for that first set of tools, and for many others that
I contributed later. He was very nice about it, and never for a minute, over
many years, did he stop that.

**Rob:** *That’s what I would have expected of Vaughan. But why
didn’t you want to work on the new polynomial?*

**Joan:** You are asking good questions, Rob. Basically, I did not
have the time for it. I was in the middle of work with Caroline Series that
interested me deeply. Vaughan was extremely excited and he was studying
the literature on links and knots
and braids nonstop. I understood by lunchtime that day that there was no
way that he would have been willing or able to wait for me to catch up to him.
So I made a snap decision, and, yes, I had some small regrets, but basically
I was OK with my decision.

Series and I were trying to prove, and eventually did [15], the following theorem about what is now called the Birman–Series sets:

Let __\( F_g, g > 1 \)__ be a closed surface or
a closed surface minus a finite set of points. Then __\( S_k \)__, the set of
points in __\( F_g \)__ which lie on some geodesic in __\( G_k \)__, is nowhere dense and
has Hausdorff dimension one.

I was on my way to visit her, and we had planned two quiet weeks of working together, and were hoping to finish our paper, and suddenly a new knot polynomial had appeared on the scene. I thought, either I’m going to drop what I’m doing with Caroline and study nonstop to try to catch up with Vaughan, or I must let Vaughan take over. He clearly wants to tell the world about it, but I don’t want to do that work now, I just don’t have the time for it.

**Rob:** *Yes. That’s a real dilemma.*

**Joan:** My decision was that I would just go on
and do my own work.
I never really regretted what I did.

In fact, my contributions continued over the entire year after the initial discovery, and were substantially more than they had been at our second meeting. Vaughan and I had an extensive correspondence, and you can see it in the letters he wrote to me.2 He kept asking me questions that were natural and appropriate. For example, when it came to a representation of the mapping class group of a surface I knew exactly what to tell him to look for, and the way to find it, and he did find it.

There was another major piece of his initial paper that came from my work, and it had to do with what he called the “powers trace” and plat representations of knots and links. I understood the difference between the representations of links as closed braids and as plats, in the former case the invariant was defined on conjugacy classes, but in the latter case it was defined on certain double cosets in the braid group. Even more, I had proved the analogue of the Markov Theorem on double cosets. So when Vaughan told me, in the letters, that there was another trace on the matrices that he’s looking at (the Hecke algebra representations), I understood immediately that his second trace came from double cosets in the braid group, rather than from conjugacy classes. I had studied plat representations (bridge representations) of knots and links and I knew that if you closed a braid with bridges, you also got all knots and links. So I knew just where to go to, and I told him this; so one thing after another, like that. And by the time I came back from Warwick, he was already on his way to Berkeley and all this developed over that year.

**Rob:** *Right. So,
I would have said that the thing that what you missed
was the HOMFLY-PT 2-variable polynomial, because you probably
would have thought of that also if you’d been paying attention.
In the summer of 1984 I was in Cambridge (England) and first heard about the
Jones polynomial from*
*Ken Millett*.
*He and*
*Raymond Lickorish*
*were working out
a 2-variable polynomial. Later we found out that*
*Jim Hoste*, *Adrian Ocneanu**,* *Peter Freyd*
*and*
*David Yetter*, *and*
*Józef Przytycki*
*and*
*Paweł Traczyk**,* *had independently and with varied methods, also found the same polynomial.
HOMFLY-PT is an acronym from their last names, with the PT sometimes missing
because news of their work in Poland was transmitted late to the West.*

**Joan:** Rob, I never wanted to work on the HOMFLY-PT polynomial. That
aspect of knot theory just did not interest me very much. Even more,
there is a piece of the story of the HOMFLY-PT polynomial that may not be
generally known. At our second meeting, in my office, the day that the
Jones polynomial came into existence, Jones had told me that the Hecke
algebra representations are two-row representations of __\( B_n \)__, but there
was also a two-variable algebra,
that was a lift of the full representation of the symmetric group, so there
was probably a 2-variable polynomial. He said that right away, and six months
later Ocneanu gave that proof of the existence of the HOMFLY-PT polynomial.

**Rob:** *Well, that’s interesting. Yet he did not work that
out himself.*

**Joan:** No, he probably didn’t have the time for it. He was getting
ready to participate in the special year at the Mathematical Sciences
Research Institute (MSRI), which by an incredible
coincidence was dedicated to (1) operator algebras and (2) knot theory,
and was eager to get as much of the hard work done as possible before going
to MSRI for the year.

**Rob:** *Right.*

**Joan:** And I have here in my book, that I went to Berkeley for a
workshop at MSRI, October 10-16, 1984.

**Rob:** *That would have been an introductory workshop.*

**Joan:** Vaughan told me that he was giving lectures on operator
algebras and knots and that he was disappointed
at the small audience he had.

[Both laugh.]

**Joan:** So it didn’t catch on immediately with the knot theorists.

**Rob:** *Well it did, it did that summer of ‘84, because
the various people who got the HOMFLY-PT, there’s five groups there, and
they caught on. But very few of the knot theorists wanted to take up
operator algebras.*

**Joan:** I should add that I wrote one short paper, during that
first summer
[14].
The work in that paper answered one of my
questions — was this actually a complete knot invariant? — and I proved
the answer was a resounding “no”.

**Rob:** *Right. But it definitely has been a gold mine
for mathematics and it did answer some old questions in
knot theory, such as one of the classical Tait conjectures, that any
reduced (no nugatory crossings) diagram of an alternating link has the
fewest possible crossings. But I’ve often felt that it wasn’t immediately
a gold mine for topology because
it was not discovered in order to solve topological problems. It was
discovered because Vaughan was doing operator algebras,
and then his work had this surprising application. The point is that it
wasn’t designed for topology. Later the Jones polynomial was categorized
by*
*Khovanov* *and this determines the unknot, so the impact of the Jones
polynomial grew and continues to grow.*

**Joan:** I disagree with part of what you say, namely “it was
not discovered in order to solve topological problems”. To Vaughan,
the trace function seemed magical in some ways. However, for example,
when you realize the trace it is not changed by exchange moves on braids,
you understand the apparent magic is very natural in a topological setting.

**Rob:** *What else? You wrote papers with*
*Hans Wenzl*.
*How did that come about?*

**Joan:**
After the discovery of the two-variable polynomial,
Louis Kauffman
gave his own very elegant proof of the existence of the 1-variable Jones polynomial,
using what became known as *diagrammatic methods*. You surely know
that paper, it’s a gem. Lou then went on and did more, using diagrammatic
methods to prove the existence of yet another knot polynomial, dubbed the
*Kauffman polynomial*. I felt that the Kauffman polynomial, like the
Jones polynomial, should come from a
trace on an algebra. So I suggested this to Wenzl who was at the
time a graduate student working with Vaughan at Penn, and we started to
talk. Ultimately, we found the appropriate algebra and a trace function
on this new
algebra was, of course, the Kauffman polynomial, because the algebra had
been designed with that in mind.

**Rob:** *Do you remember what year you got together and worked
with Hans?*

**Joan:** As I recall it, Wenzl gave the first talk on our joint work
at the Santa Cruz Workshop on “Braids”, held the summer of 1985.
As it happens, the same idea had occurred, simultaneously, to
Jun Murakami and he discovered the BMW algebra independently. While our two approaches
(i.e., Murakami’s and mine with Wenzl) were different, the algebra became
known for the three of us, even though we neither worked together nor
discussed our work until many years later.

#### Vassiliev invariants

**Rob:** *So let’s talk about
Vassiliev.
How did you get
started? What was your entrée into Vassiliev invariants?*

**Joan:** Some time around 1990
Arnol’d
came to
visit the Columbia Mathematics Department. I arrived one day early in the
semester and
as I came into the Columbia math building there was Arnold lugging a huge
suitcase with several black belts wrapped around it to keep it from opening.

**Rob:** *Yes.* [laughs]

**Joan:** He came to Columbia right from the airport. So I said to him,
“Well, I want to go down and collect my mail,” which was half a flight
down, whereas math offices began half a flight up, where the elevator
was located.
He followed me to the mailbox and opened his suitcase, with all its belts,
right there
on the floor. Arnold was a very charismatic, lively guy and was often
brimming with excitement (although he could also be quite
morose) in those days.
He said, “Oh, I want to talk to you today because I
have some work from my student Vassiliev who has discovered lots of new
knot invariants.”
Thinking of the Jones and HOMFLY and Kauffman polynomials and their many
relatives, my first
reaction was “please, no more knot invariants!”.

**Rob:** [laughs]

**Joan:** However, as it turned out, Vassiliev’s invariants were
immediately appealing to me because (unlike the Jones polynomial,
which was at heart a combinatorial object) they were grounded in classical
topology.
Vassiliev considers the space of all knots, which he thinks of as the space of
all smooth embeddings __\( \mathbb R \to \mathbb R^3 \)__ which are asymptotically
close to the standard embedding near __\( \pm \infty \)__, i.e., the space of all
“long knots”. It’s a subspace of the space __\( \mathcal M \)__ of all smooth
*maps* __\( \mathbb R \to \mathbb R^3 \)__ where again the behavior has to be
correct near __\( \pm\infty \)__. The latter space is divided into “chambers” by
walls, the *discriminant* __\( \Sigma \)__, which are penetrated (in Vassiliev’s
picture) when one makes crossing changes. He fixes the instant when there
is a transverse double point, where in general many such double points will
be needed to pass from one
chamber to another. He uses known techniques
to study the cohomology of
__\( \mathcal M \setminus \Sigma \)__. I regarded
his work as being solidly grounded in topology.

Xiao-Song Lin (at that time a Ritt Assistant Professor at Columbia) and I sat down together at tea with Arnold. What Arnold wanted us to do initially was just to mail out copies of Vassiliev’s paper [e7] to all the knot theorists that I knew in the United States. Lin and I took on this tedious job. At that time you had to stand and feed pages into the Xerox machine, one at a time. When it got too hot, it stopped working. So finally we got enough copies together and stapled them and addressed all the envelopes and sent them out. All that took us quite some time, but of course, while we were doing the copying we started to talk about the new invariants. That was how our collaboration began. My intuition was that Vassiliev invariants were closely related to the Jones polynomial and its relatives, and the possibility that my guess might be true interested Lin. Our joint work was aimed at making sense of that guess.

**Rob:** How did you approach that matter?

**Joan** We knew that the Jones polynomial could be characterized by
a set of axioms. So our initial steps were to try to do
the same for Vassiliev invariants. We succeeded in doing that, but it did
not immediately suggest to us
what the relationship should be. After we had the axioms
Xiao-Song gave a talk at the Institute.
Ed Witten
was in the audience and
came over and spoke to
Xiao-Song afterwards and said, “My student
Bar-Natan is doing some work
that sounds like it’s pretty closely related to what
you and Joan are you thinking about.” The work Dror had done before we
began our discussions involved Feynman diagrams, and came out of mathematical
physics. At Witten’s suggestion, Dror Bar-Natan called us and we arranged
the first of several discussions, all at Columbia. We soon realized
that what Dror had been working on was the very simplest case of
Vassiliev invariants; the polynomial that came out of our axioms, in
that case,
was the ubiquitous Alexander polynomial.

At first, we did not know how to pass from the Alexander polynomial to the
Jones polynomial and its relatives.
Then Xiao-Song
said, to me, “I have an idea.” This really was *his* idea, although it
was certainly motivated by both the axioms that he and I had developed
together, and Dror’s work on the Alexander polynomial. His idea was that
the coefficients in other power series, chosen with the Jones polynomial
in mind, would also
turn out to be Vassiliev invariants.

I want to interrupt our discussion for a moment, Rob, to discuss traditions in mathematics regarding joint work. My experience is that authors are always listed alphabetically, without questions being asked as to which one contributed this part or that part of the work. It carries over to papers published by graduate students. While the advisor inevitably plays a large role in a graduate student’s thesis, that role does not carry over to putting the advisor’s name as a coauthor in a thesis. I respect and value that tradition, and prefer it to all other ways of dividing credit. I recall being on an “ad hoc review committee” for an appointment in Physics, where the candidate was always one of 50 or more authors, in every one of his papers. How could we know what he/she contributed? I note that in mathematics there is a companion tradition that the Field’s Medal is awarded to young mathematicians, and that tradition has lead to deep respect in the community for the work of the youngest mathematicians.

Returning to the matter at hand, the very next day Lin explained his idea to me. We both understood immediately that the connection we had been seeking between the Jones polynomial and Vassiliev invariants, and through them topology, had been established. We soon generalized the connection that had been made to the HOMFLY and Kauffman polynomials.

Dror was not entirely happy about that. He said, “Well, you know, I have
to write a
thesis and I think I want to do this by myself.” So he didn’t want to work
with us! [*laughs*] I thought it was quite unfortunate that the
three of us had not written a paper together,
and while I understood his reasons, it seemed to me that the overlap of
our joint work with his earlier work was minimal. However, rather than
discuss how to divide things up, Dror insisted on a complete split, and
that’s what happened.
His paper
[e9]
used our axioms to define Vassiliev invariants.
He called them
*finite-type* invariants.
The word finite-type came out of our axioms. This meant that he bypassed the
hard work and insights that we had put into the axioms with his definitions.
He then used finite-type invariants to establish a key connection with the
Reshetikhin–Turaev invariant. All this is explained very carefully
in
Simon Willerton’s excellent review for *Mathematical Reviews* (see
[e9]).
Dror’s very appealing and beautifully written paper came to be regarded
as the standard introduction to Vassiliev invariants. It was partially
expository, and he presented our work as part of it, but it was also rich
in new ideas. It lead the reader, gently, into the study of the Kontsevich
integral. In fact, the first real proof of the validity of the Kontsevich
integral is in Dror’s paper. The topological origins fell by the wayside.

Our paper [23] became available in the math community more or less simultaneously with Dror’s paper [e9], but it was used by others primarily in the service of what eventually became known as topological quantum field theory.

**Rob:** *I see.*

**Joan:** In our paper we developed the axioms, showing they
characterized the same set of invariants as Vassiliev’s original work.
We then gave our main application: to prove that if you expanded the Jones,
HOMFLY-PT and Kauffman polynomials in power series, in a particular way,
then the coefficients in those series were Vassiliev invariants. Thus the
Jones, HOMFLY and Kauffman polynomials were generating functions for
certain infinite sequences of Vassiliev invariants.

**Rob:** *What have we learned from Vassiliev invariants? When we
were talking earlier you suggested that we haven’t really studied them.*

**Joan:** Yes, the abstract to
[23]
says, in full:

A fundamental relationship is established between Jones’ knot invariants and Vassiliev’s knot invariants. Since Vassiliev’s knot invariants have a firm grounding in classical topology, one obtains as a result a first step in understanding the Jones polynomial by topological methods.

The next step in that regard. was done by
Ted Stanford,
a graduate student
at Columbia. His PhD thesis
[e11],
written at the time we are
discussing, extended Vassiliev invariants from knots to links and certain
knotted graphs. But he also did a second piece of work
[e8]
that
related to the problem of interpreting the Jones polynomial topologically,
proving a very interesting theorem. To explain it, let __\( B_k \)__ be Artin’s
braid group and let __\( P_k \)__ be its “pure braid subgroup”, that is the
kernel of the natural homomorphism from __\( B_k \)__ to the symmetric group.
Let __\( \gamma_n(P_k) \)__ be the __\( n \)__-th group of the lower central series
of the pure braid group __\( P_k \)__. Let __\( K_1, K_2 \)__ be knots in __\( \mathbb S^3 \)__.
Here is Stanford’s theorem:

__\( v(K_1) = v(K_2) \)__ for every Vassiliev invariant of order __\( < n \)__ if and only
if there exists a positive integer __\( k \)__ and braids __\( p,b\in B_k \)__ such that
__\[ K_1=\operatorname{closure}(b), \quad
K_2=\operatorname{closure}(pb) \quad\text{and}\quad
p\in \gamma_n(P_k) .\]__

It was a fine paper, and in fact it was accepted (modulo some rewriting) by a top journal. But at the same time, the math community was much more interested in Reshetikhin–Turaev and topological quantum field theory, and Stanford was discouraged by that and put off the needed revisions. By the time they were ready, it was so long after acceptance that his paper remains an unpublished preprint [e8] to this day!

In a different direction I mention a paper by Eisermann [e14] about the Jones polynomial of ribbon knots. I feel that the topological meaning of the Jones polynomial is a problem that is within reach, but has not really grabbed the interest of enough mathematicians to make it seem like a solvable problem.

##### Editor’s Note

*The undated letters with
Dennis Johnson
included in the list of references below have been provisionally ordered
based on internal evidence and on Birman’s memory of her
correspondence with Johnson.
Shigeyuki Morita’s
evaluative help in
this effort is greatly appreciated.
A full catalog of that correspondence can be found
here.
*