Celebratio Mathematica

Joan S. Birman

An interview with Joan Birman about her mathematics

by Rob Kirby

This in­ter­view is an ed­ited tran­script of a re­cor­ded Skype in­ter­view that took place in the spring of 2018. The con­ver­sa­tion touches on the ori­gins of Joan Birman’s in­terest in braids and knots; her friend­ship and math­em­at­ic­al dis­cus­sions with Den­nis John­son and Vaughan Jones; her col­lab­or­a­tions with oth­er math­em­aticians (Bob Craggs, Car­oline Series, Bob Wil­li­ams); and the de­cisions that shaped her math­em­at­ic­al re­search.
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.


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.


[1] J. S. Birman and R. Craggs: “On the \( \mu \)-in­vari­ant of \( Z \)-ho­mo­logy 3-spheres,” Bull. Am. Math. Soc. 82 : 2 (March 1976), pp. 253–​255. MR 0397734 Zbl 0343.​55001 article

[2] D. John­son: Let­ter to J. Birman, un­dated. About sim­pli­fic­a­tions to proofs (based on phone call with Joan). misc

[3] D. John­son: Let­ter to J. Birman, un­dated. About pa­per identi­fy­ing the ker­nel of one of “Joan’s” ho­mo­morph­isms. misc

[4] D. John­son: Let­ter to J. Birman, un­dated. About pa­per enu­mer­at­ing \( \mathbb{Z}_2 \) maps, proof that all the 4-in­ter­sec­tion cases re­duce, and ma­ter­i­al on in­ter­sec­tion the­ory. misc

[5] D. John­son: Let­ter to J. Birman, un­dated. About us­ing re­la­tions in \( \mathcal{I} \) to get sym­met­ric ho­mo­logy spheres and new pa­per with de­scrip­tion of \( \mathcal{I}/\mathcal{C} \). misc

[6] D. John­son: Let­ter to J. Birman of 10 March 1977. Short de­scrip­tion of ma­chinery to be used in forth­com­ing pa­per. misc

[7] J. S. Birman and R. Craggs: “The \( \mu \)-in­vari­ant of 3-man­i­folds and cer­tain struc­tur­al prop­er­ties of the group of homeo­morph­isms of a closed, ori­ented 2-man­i­fold,” Trans. Am. Math. Soc. 237 (March 1978), pp. 283–​309. MR 0482765 Zbl 0383.​57006 article

[8] D. John­son: Let­ter to J. Birman, un­dated. About pa­per on tor­sion of maps in \( \mathcal{I} \). misc

[9] D. John­son: Notes for J. Birman, un­dated. About the space of Cas­son ho­mo­morph­isms for a sur­face \( K_{g,1} \). misc

[10] J. S. Birman and R. F. Wil­li­ams: “Knot­ted peri­od­ic or­bits in dy­nam­ic­al sys­tems, I: Lorenz’s equa­tions,” To­po­logy 22 : 1 (1983), pp. 47–​82. Part II was pub­lished in Low-di­men­sion­al to­po­logy (1983). MR 682059 Zbl 0507.​58038 article

[11]V. Jones: Let­ter to J. Birman of 31 May 1984. As a follow-up to their May 22, 1984 meeting, Jones explains to Birman, who was not familiar with his work on type \( \text{II}_1 \) factors, how that work had lead him to a formula for a 1-variable polynomial invariant of a classical link in \( \mathbb{R}^3 \). He calls his invariant \( V (t) \). Starting on page 5, he works out some of its basic elementary properties. misc

[12]V. Jones: Let­ter to J. Birman of 14 Novem­ber 1984. Birman and Jones had met at a conference at MSRI October 10–16, and discussed, among other things, forming knots and links from braids, but using the connections needed to get plat and bridge presentations. misc

[13]V. Jones: Let­ter to J. Birman of 21 Novem­ber 1984. About another topic that had been discussed at the October 10–16 gathering, i.e., representations of the mapping class group of a surface of genus 2, using 6-plats. misc

[14] J. S. Birman: “On the Jones poly­no­mi­al of closed 3-braids,” In­vent. Math. 81 : 2 (June 1985), pp. 287–​294. MR 799267 Zbl 0588.​57005 article

[15] J. S. Birman and C. Series: “Geodesics with bounded in­ter­sec­tion num­ber on sur­faces are sparsely dis­trib­uted,” To­po­logy 24 : 2 (1985), pp. 217–​225. MR 793185 Zbl 0568.​57006 article

[16]V. Jones: Let­ter to J. Birman of 26 Feb­ru­ary 1985. About the for­mula for closed 3-braids that are knots. misc

[17]V. Jones: Let­ter to J. Birman, un­dated. About matrices in \( \mathrm{SL}(5,\mathbb{R}) \). misc

[18]V. Jones: Let­ter to J. Birman of 15 May 1985. About his observation that the plat representation of the 1-variable Jones polynomial satisfies a skein relation. misc

[19]V. Jones: Let­ter to J. Birman of 31 Janu­ary 1986. A letter that told Birman about the submission of the “first draft” of “Hecke algebra representations of braid groups and link polynomials” for publication. Essentially everything that had been discussed in the letters that preceded this one (and more) appeared in the published paper. misc

[20]V. Jones: Copy of Let­ter to L. Kauff­man of 3 Oc­to­ber 1986. About a states model for the two-variable Jones polynomial. misc

[21]V. Jones: Email to J. Birman of 12 June 1990. An e-mail from V Jones to J. Birman, about calculating the braid index of a knot. To understand its content, note that near the end of Jones’ paper “Hecke algebra representations of braid groups and link polynomials”, there is a table that assigns braid indices to the 84 knots from the table at the end of Rolfsen’s classic book Knots and Links. Birman had asked Jones whether he discovered new tricks for changing knots into braids, and if not, how he had the patience to do it on so many knots? Read this 12 June 1990 e-mail to learn his answer. misc

[22] J. S. Birman: “New points of view in knot the­ory,” Bull. Am. Math. Soc. (N.S.) 28 : 2 (1993), pp. 253–​287. MR 1191478 Zbl 0785.​57001 article

[23] J. S. Birman and X.-S. Lin: “Knot poly­no­mi­als and Vassiliev’s in­vari­ants,” In­vent. Math. 111 : 2 (1993), pp. 225–​270. MR 1198809 Zbl 0812.​57011 article

[24] J. S. Birman, D. John­son, and A. Put­man: “Sym­plect­ic Hee­gaard split­tings and linked abeli­an groups,” pp. 135–​220 in Groups of dif­feo­morph­isms: In hon­or of Shi­gey­uki Mor­ita on the oc­ca­sion of his 60th birth­day (Tokyo, 11–15 Septem­ber 2006). Edi­ted by R. C. Pen­ner. Ad­vanced Stud­ies in Pure Math­em­at­ics 52. Math­em­at­ic­al So­ci­ety of Ja­pan (Tokyo), 2008. MR 2509710 Zbl 1170.​57018 ArXiv 0712.​2104 incollection