Celebratio Mathematica

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”.

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.

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.

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:

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.

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:

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.

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.