Celebratio Mathematica

Joan S. Birman

Interview with Joan Birman

by Allyn Jackson and Lisa Traynor

Joan S. Birman is a lead­ing to­po­lo­gist and one of the world’s fore­most ex­perts in braid and knot the­ory. She was born on May 30, 1927, in New York City. She re­ceived a B.A. de­gree in math­em­at­ics in 1948 from Barn­ard Col­lege and an M.A. de­gree in phys­ics two years later from Columbia Uni­versity. She worked on math­em­at­ic­al prob­lems in in­dustry for sev­er­al years, raised three chil­dren, and even­tu­ally re­turned to gradu­ate school in math­em­at­ics. She re­ceived her Ph.D. in 1968 at the Cour­ant In­sti­tute at New York Uni­versity, un­der the dir­ec­tion of Wil­helm Mag­nus. She was on the fac­ulty of the Stevens In­sti­tute of Tech­no­logy (1968–1973), dur­ing which time she also held a vis­it­ing po­s­i­tion at Prin­ceton Uni­versity. Her in­flu­en­tial book Braids, Links, and Map­ping Class Groups (An­nals of Math­em­at­ics Stud­ies, num­ber 82, 1974) is based on a series of lec­tures she gave dur­ing her time at Prin­ceton. In 1973 she joined the fac­ulty of Barn­ard Col­lege, Columbia Uni­versity, where she has re­mained ever since and where she is now Re­search Pro­fess­or Emer­it­us.

Birman’s hon­ors in­clude a Sloan Found­a­tion Fel­low­ship (1974–1976), a Gug­gen­heim Fel­low­ship (1994–1995), and the Chauven­et Prize of the Math­em­at­ic­al As­so­ci­ation of Amer­ica (1996). She was a mem­ber of the In­sti­tute for Ad­vanced Study, Prin­ceton, in spring 1987. In 1997 she re­ceived an hon­or­ary doc­tor­ate from Tech­nion Is­rael In­sti­tute of Tech­no­logy. She re­ceived the New York City May­or’s Award for Ex­cel­lence in Sci­ence and Tech­no­logy in 2005.

Birman has had twenty-one doc­tor­al stu­dents and nu­mer­ous col­lab­or­at­ors. She has served on the ed­it­or­i­al boards of sev­er­al journ­als and was among the found­ing ed­it­ors of two journ­als, Geo­metry and To­po­logy and Al­geb­ra­ic and Geo­met­ric To­po­logy. Joan Birman Both journ­als are now pub­lished by the non­profit Math­em­at­ic­al Sci­ences Pub­lish­ing Com­pany, for which Birman serves on the board of dir­ect­ors.

In 1990 Birman donated funds to the AMS for the es­tab­lish­ment of a prize in memory of her sis­ter, Ruth Lyttle Sat­ter, who was a plant physiolo­gist. The AMS Ruth Lyttle Sat­ter Prize hon­ors Sat­ter’s com­mit­ment to re­search and to en­cour­aging wo­men in sci­ence. It is awar­ded every oth­er year to a wo­man who has made an out­stand­ing con­tri­bu­tion to math­em­at­ics re­search.

What fol­lows is an ed­ited ver­sion of an in­ter­view with Joan Birman, con­duc­ted in May 2006 by No­tices Deputy Ed­it­or Allyn Jack­son and As­so­ci­ate Ed­it­or Lisa Traynor.

Early years

No­tices: Let’s start at the be­gin­ning of your life. Were your par­ents Amer­ic­an? Were they im­mig­rants?

Birman: My fath­er was born in Rus­sia. He grew up in Liv­er­pool, Eng­land, and came to the United States when he was sev­en­teen, to search for lost re­l­at­ives and to seek a bet­ter life. My moth­er was born in New York, but her par­ents were im­mig­rants from Rus­sia–Po­land.

No­tices: What did your fath­er do?

Birman: He star­ted as a ship­ping clerk in the dress in­dustry and worked his way up to be­come a suc­cess­ful dress man­u­fac­turer. He told his four daugh­ters re­peatedly that the U.S. was the best coun­try in the world, a land of op­por­tun­ity. Para­dox­ic­ally, he also told them, “do any­thing but go in­to busi­ness.” He wanted us all to study.

No­tices: Did your moth­er have a pro­fes­sion?

Birman: No, she was a house­wife. Neither of my par­ents fin­ished high school.

No­tices: Why did they em­phas­ize their four daugh­ters get­ting an edu­ca­tion?

Birman: Jew­ish cul­ture, as it was handed down to us, in­cluded the strong be­lief that Jews sur­vived for so many years in the Di­a­spora be­cause they were “the people of the book”. The free trans­la­tion, when I brought home an ex­am with a grade of 98, was “what happened to the oth­er 2 points?” Be­com­ing an edu­cated per­son, and us­ing that edu­ca­tion to do something big­ger than just to earn money, was set up to my gen­er­a­tion as a very im­port­ant goal.

No­tices: When you were a child, did you like math­em­at­ics?

Birman: Yes, I liked math, from ele­ment­ary school, and even earli­er than that, al­though I did not know enough to pin­point what I liked.

No­tices: Were there teach­ers in your early years who en­cour­aged you in math­em­at­ics, or who were in­spir­ing?

Birman: In ele­ment­ary school that’s hard to say, al­though we cer­tainly had chal­len­ging math. I went to an all-girls high school in New York, Ju­lia Rich­mond High School. It was really a rough in­ner-city high school, but with­in it there was a small aca­dem­ic unit, a school with­in a school. We had some very good teach­ers. We had a course in Eu­c­lidean geo­metry, and every single night we would have tele­phone con­ver­sa­tions and ar­gue over the solu­tions to the geo­metry prob­lems. That was my in­tro­duc­tion to proof, and I just loved it, it was won­der­ful. When the course ended, I joined a small group of girls who cam­paigned for more geo­metry, but the teach­er (her name was Miss Ma­honey) was will­ing but per­haps not know­ledge­able enough to know how to con­tin­ue to chal­lenge the in­tel­lec­tu­al in­terests of this eager group of girls! She taught us 3-di­men­sion­al Eu­c­lidean geo­metry, and that was a little dull. If she had taught us hy­per­bol­ic geo­metry, or group the­ory, where we would have en­countered new ideas, we would have been in heav­en!

No­tices: Usu­ally high school girls are on the phone talk­ing about their hair.

Birman: We did that too! Ac­tu­ally I was in this little group, and we were def­in­itely re­garded as be­ing nerds. Most of the girls in our se­lect­ive school with­in a school worked hard and got good grades, but talked all the time about boys and clothes. I was a late de­veloper and wasn’t ready for that. I didn’t date at all un­til I was in col­lege. Still, at one point I was elec­ted pres­id­ent of the class, so the oth­er stu­dents could not have been really hos­tile. I felt ac­cep­ted, and even liked. There was an at­mo­sphere of tol­er­ance.

No­tices: Were your sis­ters also in­ter­ested in math?

Birman: Yes. My old­est sis­ter, Helen, was a math ma­jor at Barn­ard, and the next one, Ruth, was a phys­ics ma­jor. Ruth ul­ti­mately be­came a plant physiolo­gist. She was Ruth Sat­ter of the Sat­ter Prize. She had a fine aca­dem­ic ca­reer, be­fore her un­timely death from leuk­emia. Helen is in­de­pend­ently wealthy and is a phil­an­throp­ist, with very spe­cial in­terests of her own. My young­er sis­ter, Ada, be­came a kinder­garten teach­er. She was less ori­ented to­ward aca­dem­ics.

No­tices: Did you like math when you went to col­lege?

Birman: Two things changed. First, the col­lege math course that I was ad­vised to take at Swarth­more was a cook­book cal­cu­lus course, and it was both bor­ing and un­con­vin­cing. So I looked around and found oth­er things that ap­pealed to me (as­tro­nomy, lit­er­at­ure, psy­cho­logy), al­though I did ma­jor in math. Then I trans­ferred to Barn­ard Col­lege, in or­der to be able to live in New York. At Barn­ard, the math of­fer­ings were all low-level. When you got to the point where you were ready for ser­i­ous math, you were dir­ec­ted to courses at Columbia, which at that time was an all-male school. That was the first time that I hit a situ­ation where I was one of a very small num­ber of girls. Most of the Barn­ard wo­men were cowed by it and gave up. Even­tu­ally I was the only girl in my classes, and I caught the idea that maybe math was not for girls.

From bachelor’s degree to industry

Birman: Yes. But there was a long gap be­fore I went on to gradu­ate school. The so­cial at­mo­sphere had presen­ted un­ex­pec­ted dif­fi­culties. My par­ents not only ex­pec­ted their four daugh­ters to get mar­ried, but we were to get mar­ried in or­der! There was all kinds of non­sense like that. But on the oth­er hand, the only way that a re­spect­able girl could get out from un­der her par­ents con­trol was to marry, so I was not averse to the idea. But I did not want to make a mis­take in my choice, and that took at­ten­tion. I did think about go­ing to gradu­ate school, but I un­der­stood how hard math was. I thought it would take lots of con­cen­trated ef­fort, as it must for any ser­i­ous stu­dent. I was afraid that I would wreck my life if I gave math that kind of at­ten­tion at that time. (I think I was right. As we talk, Joe and I have been mar­ried for fifty-six years, and he has been my biggest sup­port­er.) Ac­tu­ally, I didn’t really de­cide not to go to gradu­ate school, but when the op­por­tun­ity arose to put it off and ac­cept an in­ter­est­ing job, the job was ap­peal­ing.

The job was very nice. I was ex­tremely lucky. It was at an en­gin­eer­ing firm that made mi­crowave fre­quency meters. These meters were cyl­indric­al cans with two para­met­ers, the ra­di­us of the base and the depth (or height). The ra­di­us was fixed, but the depth could be changed with a plun­ger, chan­ging the res­on­ant fre­quency. The (depth-to-res­on­ant fre­quency) curve was non­lin­ear, and the prob­lem was that they had a hard time cal­ib­rat­ing the di­als, put­ting the notches on to in­dic­ate what the fre­quency would be as you pushed the plun­ger in. They hired me be­cause they had the idea that they could sell more meters if they could push in the plun­ger in a nov­el way that would yield an ap­prox­im­ately lin­ear re­sponse curve. In cal­cu­lus I had learned about lad­ders slid­ing against a wall, and in the job in­ter­view the idea came up that the curve that gave the height of the lad­der as a func­tion of its dis­tance from the wall might be a curve that could be fit­ted to the ex­per­i­ment­al data. The idea worked very well. For about eight months I com­puted the para­met­ers, and they con­struc­ted meters of all sizes with plun­gers that pushed in along an ax­is or­tho­gon­al to the ax­is of the can. The di­als were for all prac­tic­al pur­poses lin­ear. I was very happy!

But when that pro­ject ended, they set me to work tak­ing meas­ure­ments on an os­cil­lo­scope, and that was pretty dull. One day I happened to run in­to my old phys­ics pro­fess­or from Barn­ard, and he offered me a po­s­i­tion as the phys­ics lab as­sist­ant at Barn­ard. I took the po­s­i­tion and ap­plied to gradu­ate school in phys­ics. I real­ized that my job pos­sib­il­it­ies would im­prove if I had a phys­ics de­gree. I did get a mas­ter’s de­gree in phys­ics, but I do not have good in­tu­ition for the sub­ject. I felt they could just tell me any­thing, and I would have to be­lieve it. I am as­ton­ished these days at the way in which phys­ics has fed in­to math. Phys­i­cists do seem to have an in­tu­ition that goes bey­ond what math­em­aticians very of­ten see, and they have dif­fer­ent tests of truth. I just didn’t have that in­tu­ition. Yet I really en­joyed the phys­ics lab, be­cause when I saw things in the lab, I knew they were true. But I didn’t al­ways trust the laws of phys­ics that we learned. On the oth­er hand, I got an MA, and then I got a bet­ter job.

No­tices: This was in the air­craft in­dustry?

Birman: Yes. It was in the days of ana­log com­puters. I worked on a nav­ig­a­tion com­puter. The pi­lot would be fly­ing a plane, and the com­puter would send a radar sig­nal to the ground. The sig­nal would be bounced back to the plane. The com­puter meas­ured the Dop­pler shift and used it to com­pute air speed and alti­tude. My part of the whole thing was er­ror ana­lys­is — to fig­ure out the er­rors when the plane was be­ing bounced around by changes in air pres­sure. A second prob­lem was that of max­im­iz­ing air­craft range for a fixed amount of fuel. A third was the design of a col­li­sion avoid­ance sys­tem.

No­tices: Were there many wo­men?

Birman: I worked at three dif­fer­ent en­gin­eer­ing firms. At one of them there were sev­er­al wo­men, but at the oth­ers I don’t re­call any oth­er wo­men.

Wandering toward graduate school

No­tices: You got mar­ried when you were study­ing phys­ics in gradu­ate school. Did you stop work­ing then?

Birman: No, I con­tin­ued to work un­til I had a child, five years later. When my first child was born, I planned to go back to work be­cause I really liked what I was do­ing. But that posed a prob­lem. In those days, there was no day care. Un­less you had a fam­ily mem­ber to take care of your chil­dren (and my moth­er and moth­er-in-law were un­able to do that), it was al­most im­possible. My hus­band and I had thought, very un­real­ist­ic­ally, we will put an ad in the pa­per and hire some­body. But then, I had this huge re­spons­ib­il­ity for our baby, and I just couldn’t see leav­ing him with some­body about whom I knew very little. My hus­band was very en­cour­aging about my go­ing back to work. I did work a few days a week. First I worked two days a week, then one day a week after we had a second child. Just be­fore our third child was born, my hus­band had been in­vited to teach in a dis­tant city. He had been in in­dustry and was think­ing about a switch to aca­demia. Dur­ing that year, I had to stop my part-time job, but it had already dwindled down to one day a week. When I came back I knew I couldn’t work that way any­more. So I went to gradu­ate school with the idea of learn­ing some new things for when I’d go back to work. You can see that I led a very wan­der­ing and un­dir­ec­ted life! It amazes me that I got a ca­reer out of it — and it has been a really good ca­reer!

No­tices: When did you then de­cide that you would get a Ph.D.?

Birman: I star­ted grad school in math right after my young­er son was born, on Janu­ary 12, 1961. I went to New York Uni­versity, where my hus­band was on the fac­ulty, so that my tu­ition was free. NYU’s Cour­ant In­sti­tute had an ex­cel­lent part-time pro­gram, with even­ing courses that were es­sen­tially open ad­mis­sions. I took lin­ear al­gebra the first semester, and then real and com­plex ana­lys­is the fol­low­ing year. And then I de­cided I could handle two courses a year, and did.

One of the first courses I took was com­plex ana­lys­is, with Louis Niren­berg. In the first lec­ture he said, “A com­plex num­ber is a pair of real num­bers, with the fol­low­ing rules for adding and mul­tiply­ing them.” I cer­tainly knew about “ima­gin­ary num­bers”, but he put them in­to a frame­work that was sound math­em­at­ics. It sounds like a trivi­al change, but it was not. Even­tu­ally, I also had a course in to­po­logy, which I loved, with Jack Schwartz. He was not a to­po­lo­gist, and when I go back and look at my notes, I see it was a weird to­po­logy course! He was some­body who liked to try new things. He taught us co­homo­logy in a be­gin­ning to­po­logy course — not ho­mo­logy, not even the fun­da­ment­al group! But I really loved that course. It really grabbed me, al­though the ap­proach had its down side, as I knew al­most no ex­amples. I had star­ted study­ing at Cour­ant with the in­ten­tion of learn­ing some ap­plied math­em­at­ics. But everything I learned pushed me to­ward pure math­em­at­ics.

At Cour­ant I was start­ing to pile up enough courses for an MA, and there was a re­quired mas­ter’s fi­nal ex­am. When I took the ex­am, I didn’t real­ize it was also the Ph.D. qual­i­fy­ing ex­am. I was sur­prised when I passed it for the Ph.D. That’s when I ap­plied for fin­an­cial as­sist­ance, but to get it I had to be a full-time stu­dent. So that’s really when I star­ted on a Ph.D. track. There were not many wo­men around. The people in the de­part­ment were very nice to me — they real­ized that I had three chil­dren, and they did not give me heavy TA as­sign­ments. Kar­en Uh­len­beck was one of the stu­dents there, but she trans­ferred out. Cath­leen Mor­awetz was on the fac­ulty, and I took one course from her.

No­tices: Your ad­visor was Wil­helm Mag­nus. How did he end up be­ing your ad­visor?

Birman: After passing the qual­i­fy­ing ex­ams, one had to take a series of more spe­cial­ized ex­ams for ad­mis­sion to re­search. My hus­band was on the NYU fac­ulty, and the first ques­tion I was asked in one of the ex­ams was, “Who is smarter, you or your hus­band?”

No­tices: That was the first ques­tion?

Birman: Yes, it’s ludicrous, in 2006. Later on when I be­came a math­em­atician, I met the per­son who asked this ques­tion and re­minded him of it, and he said, “Oh no, not me! I didn’t say that!”

No­tices: How did you an­swer the ques­tion?

Birman: I laughed. It was the only thing to do. Af­ter­wards I star­ted to get really angry about it. It was a stu­pid ques­tion!

Any­way, I passed that ex­am too and went look­ing for an ad­visor. The first per­son I ap­proached was the to­po­lo­gist Michel Ker­vaire, but he wasn’t in­ter­ested. He said, “You’re too old and you don’t know enough to­po­logy.” He was right, I didn’t know enough to­po­logy. And I can un­der­stand why he would be skep­tic­al of a per­son my age. You have to be con­vinced when you see someone who is out­side of the usu­al frame­work that the per­son is a ser­i­ous stu­dent, and he had nev­er been my classroom teach­er.

I went to speak to Niren­berg. He was very help­ful to me. I read the No­tices in­ter­view with him, and he had told you that he loved in­equal­it­ies. That’s funny, be­cause I re­mem­ber he asked me, “Do you like in­equal­it­ies?” And I said, “No, I don’t like in­equal­it­ies!” He said, “Then you don’t want to study ap­plied math.” And he was right!

No­tices: That was a good ques­tion to ask!

Birman: It was an ex­cel­lent ques­tion. After that I went to talk to Wil­helm Mag­nus. He had no­ticed me, be­cause I had done some grad­ing for him. He was an al­geb­ra­ist, but he had no­ticed that I loved to­po­logy, and so he met me halfway and gave me a pa­per to read about braids. That showed great sens­it­iv­ity on his part. It was a ter­rif­ic top­ic. He later told me of his habit of pick­ing up strays, and in some way I was a stray.

No­tices: What pa­per was it that he gave you?

Birman: It was a pa­per by Fadell and Neuwirth [e1].

The braid groups were defined in that pa­per as the fun­da­ment­al group of a cer­tain con­fig­ur­a­tion space. Mag­nus said that he didn’t un­der­stand the defin­i­tion, and it took me a long time to un­der­stand it. Fi­nally I did, and I was very happy. Mag­nus had worked on the map­ping class group of a twice-punc­tured tor­us, and he had sug­ges­ted that I could ex­tend this work to a tor­us with 3 or 4 punc­tures. My thes­is ended up be­ing about the map­ping class group of sur­faces of any genus with any num­ber of punc­tures. He thought that was a real achieve­ment. As soon as I un­der­stood the prob­lem well enough, I solved it. It was both fun and very en­cour­aging.

Around this time there was a very dif­fer­ent pa­per by Garside on braids that in­ter­ested me greatly [e2]. I was aware of the fact that there was a scheme for clas­si­fy­ing knots with braids. When I saw that Garside had solved the con­jugacy prob­lem in the braid group, I thought that was go­ing to solve the knot prob­lem. I couldn’t have been more mis­taken, but still, it grabbed my in­terest. I am still work­ing on it — right now I am try­ing to show that Garside’s al­gorithm can be made in­to a poly­no­mi­al al­gorithm. This is im­port­ant in com­plex­ity the­ory. So my in­terest in that prob­lem dates back to gradu­ate school.

Moving into research

No­tices: After you got your Ph.D., you got a job at Stevens In­sti­tute of Tech­no­logy.

Birman: I had not done a thor­ough job on ap­plic­a­tions and was not offered any job un­til late Au­gust 1968, when Stevens In­sti­tute had some un­ex­pec­ted de­par­tures. The first year I was there I star­ted work­ing with Hugh M. Hilden (who is known as Mike). We solved a neat prob­lem that year and wrote sev­er­al really good pa­pers. The one I like best is the first in the series [2].

The work with Hilden was very re­ward­ing. My thes­is had been on the map­ping class group of a punc­tured sur­face. I showed there is a ho­mo­morph­ism from the map­ping class group of a punc­tured sur­face to that of a closed sur­face, in­duced by filling in the punc­tures. I worked out the ex­act se­quence that iden­ti­fied the ker­nel of that ho­mo­morph­ism, but I didn’t know a present­a­tion for the coker­nel, the map­ping class group of a closed sur­face, and real­ized that was a prob­lem that I would like to solve. The whole year I talked about it to Mike, whose of­fice was next to mine, and fi­nally we solved the prob­lem for the spe­cial case of genus 2. As it turned out, our solu­tion had many gen­er­al­iz­a­tions, but the key case was a closed sur­face \( \Sigma \) of genus 2. In that case, the map­ping class group has a cen­ter, and the cen­ter is gen­er­ated by the class of an in­vol­u­tion that I’ll call \( \mathcal{I} \). The or­bit space \( \Sigma/\mathcal{I} \) is a 2-sphere \( S^2 \), and the or­bit space pro­jec­tion \( \Sigma \rightarrow \Sigma/\mathcal{I}= S^2 \) gives it the struc­ture of a branched cov­er­ing space, the branch points be­ing the im­ages on \( S^2 \) of the 6 fixed points of \( \mathcal{I} \). We were able to use the fact that the map­ping class group of \( S^2_6 \) of 2 minus those 6 points was a known group (re­lated to the braid group), to find a present­a­tion for the map­ping class group \( \mathcal{M}(\Sigma) \) of \( \Sigma \). The dif­fi­culty we had to over­come was that map­ping classes are well-defined only up to iso­topy. We knew that in genus 2, every map­ping class was rep­res­en­ted by a map that com­muted with \( \mathcal{I} \), but we did not know wheth­er every iso­topy could be de­formed to a new iso­topy that com­muted with \( \mathcal{I} \). We felt it had to be true, but we couldn’t see how to prove it. One day Mike and I had the key idea, to­geth­er. The idea was to look at the path tra­versed on \( \Sigma \) by one of the 6 fixed points, say \( p \), un­der the giv­en iso­topy. This path is a closed curve on \( \Sigma \) based at \( p \). Could that closed curve rep­res­ent a non­trivi­al ele­ment in \( \pi_1(\Sigma,p) \)? It was a key ques­tion. Once we asked the right ques­tion, it was easy to prove that the an­swer was no, and as a con­sequence our giv­en iso­topy could be de­formed to one that pro­jec­ted to an iso­topy on \( S^2_6 \). As a con­sequence, there is a ho­mo­morph­ism \( \mathcal{M}(\Sigma) \rightarrow \mathcal{M}(S^2_6) \), with ker­nel \( \mathcal{I} \). Our hoped-for present­a­tion fol­lowed im­me­di­ately. It was a very fine ex­per­i­ence to work with Mike, to get to know him as a per­son via shared math­em­at­ics. It was the first time I had done joint work, and I en­joyed it so much that ever since I have been alert to new col­lab­or­a­tions. They are dif­fer­ent each time, but have al­most all been re­ward­ing.

At that point I was thor­oughly in­volved in math­em­at­ics. But my hus­band had a sab­bat­ic­al, and I had prom­ised him that I would take a year off so that he could spend his sab­bat­ic­al with col­lab­or­at­ors in France. So I took a leave of ab­sence from my job and found my­self in Par­is, and in prin­ciple it should have been a lovely year. But we had three chil­dren, and once again I had lots of home re­spons­ib­il­it­ies! Moreover, I didn’t know any of the French math­em­aticians, be­cause I had come to France without any real in­tro­duc­tions, and nobody was in­ter­ested in braids. French math­em­at­ics at that time was heav­ily in­flu­enced by the Bourbaki school. I found my­self very isol­ated and dis­cour­aged. Look­ing for a prob­lem that I could handle alone, I de­cided to do a cal­cu­la­tion.

There is a ho­mo­morph­ism from the map­ping class group of a sur­face to the sym­plect­ic group. People knew de­fin­ing re­la­tions for the sym­plect­ic group, but not for the map­ping class group, un­less the genus is \( \leq 2 \). I was in­ter­ested in the ker­nel of that ho­mo­morph­ism, which is called the Torelli group. It was an im­mense cal­cu­la­tion. I fin­ished it, and I did get an an­swer [1], which was later im­proved with the help of a Columbia gradu­ate stu­dent, Jerome Pow­ell. In 2006 a gradu­ate stu­dent at the Uni­versity of Chica­go, Andy Put­man, con­struc­ted the first con­cep­tu­al proof of the the­or­em that Pow­ell and I had proved. Put­man’s proof fi­nally veri­fies the cal­cu­la­tion I did that year in France!

When I re­turned from France I was in­vited to give a talk at Prin­ceton on the work that Hilden and I had done to­geth­er. That was when my ca­reer really began to get go­ing, be­cause people were in­ter­ested in what we had done. I was in­vited to vis­it Prin­ceton the fol­low­ing year. I did that, com­mut­ing from my home in New Rochelle, New York, to Prin­ceton, New Jer­sey. That was a very long com­mute.

No­tices: Was it around this time that you gave the lec­tures that be­came your book “Braids, Links, and Map­ping Class Groups” [3]?

Birman: Ex­actly. The lec­tures were at­ten­ded by a small but in­ter­ested group, in­clud­ing Ral­ph Fox and Kunio Mur­as­ugi, and James Can­non, at that time a postdoc. Dmitry Papakyriako­po­l­ous was also at Prin­ceton, and he was very wel­com­ing to me.

Braids had not been fash­ion­able math­em­at­ics, and their role in knot the­ory had been largely un­developed. Three top­ics that I de­veloped in the lec­tures and put in­to the book were: (1) Al­ex­an­der’s the­or­em that every link type could be rep­res­en­ted, nonu­niquely, by a closed braid, (2) Markov’s the­or­em, which de­scribed the pre­cise way in which two dis­tinct braid rep­res­ent­at­ives of the same link type were re­lated, one of those moves be­ing con­jugacy in the braid group, and (3) Garside’s solu­tion to the prob­lem of de­cid­ing wheth­er two dif­fer­ent braids be­longed to the same con­jugacy class. I had chosen those top­ics be­cause I was in­ter­ested in study­ing knots via closed braids, and to­geth­er (1), (2), and (3) yiel­ded a new set of tools.

When I had planned the lec­tures at Prin­ceton, to my dis­may I learned that there was no known proof of Markov’s the­or­em! Markov had an­nounced it in 1935, and he had sketched a proof but did not give de­tails, and the dev­il is al­ways in the de­tails. When I told my former thes­is ad­visor, Wil­helm Mag­nus, he re­marked that the sketched proof was very likely wrong! But luck­ily, I was able to fol­low Markov’s sketch, with the help of some notes that Ral­ph Fox had taken at a sem­in­ar lec­ture giv­en by a former Prin­ceton grad stu­dent (his name van­ished when he dropped out of grad school). After some num­ber of 2:00 a.m. bed­times I was able to present a proof. There are now some six or sev­en con­cep­tu­ally dif­fer­ent proofs of this the­or­em, but the one in my 1974 book was the first.

Knot polynomials and invariants

No­tices: Can you tell us about your in­ter­ac­tion with Vaughan Jones, when he was get­ting his ideas about his knot poly­no­mi­al?

Birman: One day in early May 1984, Vaughan Jones called to ask wheth­er we could get to­geth­er to talk about math­em­at­ics. He con­tac­ted me be­cause he had dis­covered cer­tain rep­res­ent­a­tions of the braid group and what he called a “very spe­cial” trace func­tion on them, and people had told him that I was the braid ex­pert and might have some ideas about its use­ful­ness. He was liv­ing in New Jer­sey at the time, so he was in the area, and we agreed to meet in my of­fice. We worked in very dif­fer­ent parts of math­em­at­ics and we had the ex­pec­ted dif­fi­culties in un­der­stand­ing each oth­er’s lan­guages. His trace arose in his work on von Neu­mann al­geb­ras, and it was re­lated to the in­dex of a type \( \mathrm{II}_1 \) sub­factor in a factor. All that was far away from braids and links. When we met, I told him about Al­ex­an­der’s the­or­em, and Markov’s the­or­em, and Garside’s work. He told me about his rep­res­ent­a­tions and about his trace func­tion. Of course, his ex­plan­a­tions were giv­en in the con­text of op­er­at­or al­geb­ras. I re­call that I said to him at one point, Is your trace a mat­rix trace? And he said no, it was not. Well, that an­swer was cor­rect, but he did not say that his trace was a weighted sum of mat­rix traces, and so I did not real­ize that, if one fixed the braid in­dex, the trace was a class in­vari­ant in the braid group. He un­der­stood that very well and did not un­der­stand what I had missed. He would will­ingly have said more, if he had, be­cause he is su­per-gen­er­ous and truly de­cent. In between our meet­ings he gave the mat­ter much thought (which I did not!), and one night he had the key idea that by a simple res­cal­ing of his trace, it would in fact be­come in­vari­ant un­der all the moves of Markov’s the­or­em, and so be­come a link in­vari­ant. He told me all this, in great ex­cite­ment, on the tele­phone. The proof that his nor­mal­ized trace was a link in­vari­ant was im­me­di­ate and crys­tal clear. After all, a good part of my book had been writ­ten with the goal of mak­ing the Al­ex­an­der and Markov the­or­ems in­to use­ful tools in knot the­ory, and Vaughan had used them in a very straight­for­ward way.

Was his new in­vari­ant really new, or a new way to look at something known? He did not know. Ex­amples were needed, and a few days later we met again, in my of­fice, to work some out. That was prob­ably May 22, 1984. The new link in­vari­ant was a Laurent poly­no­mi­al. My first thought was: it must be the Al­ex­an­der poly­no­mi­al. So I said, “Here are two knots (the tre­foil and its mir­ror im­age) that have the same Al­ex­an­der poly­no­mi­al. Let’s see if your poly­no­mi­al can dis­tin­guish them.” To my as­ton­ish­ment, it did! Well, we checked that cal­cu­la­tion very care­fully, on lots more ex­amples, be­cause the im­plic­a­tions were hard to be­lieve. By pure ac­ci­dent, I had re­cently worked out a closed braid rep­res­ent­at­ive of the Kinoshita–Ter­a­sake 11-cross­ing knot, whose Al­ex­an­der poly­no­mi­al was zero. Fish­ing it out of my file cab­in­et we learned very quickly, that same day, that the new poly­no­mi­al was nonzero on it. So in just that one af­ter­noon, we knew that he not only had a knot in­vari­ant, but even more it was brand new. I re­mem­ber cross­ing Broad­way on my way home that night and think­ing that nobody else knows this thing ex­ists! It was an amaz­ing dis­cov­ery. Very quickly, oth­er parts of the new ma­chinery came to bear, and the world of knot the­ory ex­per­i­enced an earth­quake. There was not just the Jones poly­no­mi­al, but also its cous­ins, the HOM­FLY and the Kauf­man poly­no­mi­als, and lots more. And some of the stuff in my book about map­ping class groups was rel­ev­ant too. Much later, Garside’s ma­chinery ap­peared too, in a par­tic­u­lar ir­re­du­cible rep­res­ent­a­tion of the braid group that arose via the same circle of ideas. Garside’s solu­tion to the word prob­lem was used by Daan Kram­mer to prove that braid groups are lin­ear.

There was an­oth­er re­lated part to this story. In 1991 Vladi­mir Arnold came to the United States to vis­it Columbia for a semester. I knew Arnold and met him in the lobby as he ar­rived, in Septem­ber, with his suit­case. He is a very ex­cit­able and en­thu­si­ast­ic man. He put down his suit­case right then and there and opened it on the floor next to the el­ev­at­or to get out a pa­per he had brought for me. It was by his former stu­dent Vikt­or Vassiliev. He said, “You have to read this pa­per, it’s won­der­ful, it con­tains new knot in­vari­ants, and they come from sin­gu­lar­ity the­ory, and it’s fine work, and I would like your help in pub­li­ciz­ing it!” Of course I looked at the pa­per. At that point there had been an ex­plo­sion in new knot in­vari­ants, and the open ques­tion was what they meant geo­met­ric­ally. And here Arnold was, with more in­vari­ants! The old ones were poly­no­mi­als, the new ones were in­tegers (lots of in­tegers!). Arnold asked me to copy and dis­trib­ute the pa­per in the United States. So one af­ter­noon shortly after his ar­rival I made lots of pho­to­cop­ies, and sent them out to every­one I could think of who seemed ap­pro­pri­ate. But even as I did it I sus­pec­ted the knot the­ory com­munity might not be so over­joyed to have yet more knot in­vari­ants com­ing un­ex­pec­tedly out of left field! There is res­ist­ance to learn­ing new things. We had just learned about op­er­at­or al­geb­ras, and sud­denly we had to learn about sin­gu­lar­ity the­ory! But Arnold kept after me, at tea every day.

Xiao-Song Lin was an as­sist­ant pro­fess­or in the de­part­ment, and his field is knot the­ory. We ran a sem­in­ar to­geth­er and talked every day. We were good friends, and he was al­ways ready to talk about math. I told him about the pa­per of Vassiliev. We read it to­geth­er, and we fi­nally un­der­stood most of it. We said, here are the Vassiliev in­vari­ants, and there are the knot poly­no­mi­als — and they must be re­lated in some way. But how? For a fixed knot or link, its Jones poly­no­mi­al was a one-vari­able Laurent poly­no­mi­al with in­teger coef­fi­cients, where­as its Vassiliev in­vari­ants were an in­fin­ite se­quence of in­tegers, or pos­sibly of ra­tion­al num­bers.

We had an idea that per­haps we should, for the mo­ment, set aside the fact that the Vassiliev in­vari­ants came from the ma­chinery of sin­gu­lar­ity the­ory, and try to con­struct them from their prop­er­ties. We did that be­cause we knew that the Jones poly­no­mi­al (the simplest of the knot poly­no­mi­als) could be con­struc­ted from its prop­er­ties. We thought that might be a way for us see a con­nec­tion. That had good and bad con­sequences. The bad one was that later, Vassiliev in­vari­ants were re­named “fi­nite type in­vari­ants”, and were defined via our ax­ioms. In the pro­cess their ori­gins in sin­gu­lar­ity the­ory were lost and re­main un­der­developed to this day.

Soon Lin and I real­ized how to make the con­nec­tion we had been seek­ing. We had the idea of mak­ing a change of vari­ables in the Jones poly­no­mi­al, chan­ging its vari­able from \( x \) to \( t \), with \( x = e^t \).

The Jones poly­no­mi­al was a Laurent poly­no­mi­al in \( x \), and \( e^{kt} \) has an ex­pan­sion in pos­it­ive powers of \( t \) for every pos­it­ive and neg­at­ive in­teger \( k \). This change in vari­ables changes the Jones poly­no­mi­al to an in­fin­ite series in powers of \( t \). We were able to prove that the coef­fi­cients in that in­fin­ite series sat­is­fied all of our ax­ioms for Vassiliev in­vari­ants, and so were Vassiliev in­vari­ants [5]. Everything went quickly with that idea — even­tu­ally all the knot poly­no­mi­als were re­lated to Vassiliev in­vari­ants in this way. They are gen­er­at­ing func­tions for par­tic­u­lar in­fin­ite se­quences of FT in­vari­ants. But in fact the set of FT in­vari­ants is lar­ger than those com­ing from knot poly­no­mi­als. They are more fun­da­ment­al ob­jects.

Rich problems, rich collaborations

No­tices: Can you tell us about your re­cent work with Menasco that in­volved the Markov the­or­em?

Birman: That is an­oth­er as­pect of the same un­der­ly­ing pro­ject, to un­der­stand knots through braids. In 1990 at the In­ter­na­tion­al Con­gress in Kyoto, when Vaughan Jones got the Fields Medal, I gave a talk on his work. Af­ter­ward Bill Menasco in­vited me to give a col­loqui­um based on it in the math de­part­ment in Buf­falo. So I gave a talk there about Vaughan Jones’s work, and I stayed at Bill’s house that night. We star­ted to talk, and he said, “What prob­lems are you work­ing on? What’s your dream?” I told him my dream is to clas­si­fy knots by braids. I had an idea about how you could avoid the “sta­bil­iz­a­tion” move in Markov’s the­or­em. Then about three weeks later, I got a let­ter from him say­ing “I have an idea how we might try to prove the ‘Markov the­or­em without sta­bil­iz­a­tion’ (MT­WS).” And that’s when our col­lab­or­a­tion began. Of course, my ori­gin­al con­jec­ture was much too simple. We kept solv­ing little pieces of the sought-for the­or­em. We wrote eight pa­pers to­geth­er. The last one stated and proved the MT­WS [7]. There was also an ap­plic­a­tion to con­tact to­po­logy [8].

I like to col­lab­or­ate. My col­lab­or­at­ors are also my best friends. Bill Menasco and I are very good friends. We have had such a long col­lab­or­a­tion. But we have very dif­fer­ent styles. He can sit in a chair and stare at the ceil­ing as he works on math­em­at­ics, but I like to talk about it all the time.

No­tices: Why do you do math­em­at­ics?

Birman: To put it simply, I love it. I’m re­tired right now, I don’t have any ob­lig­a­tions, and I keep right on work­ing on math. Some­times math­em­at­ics can be frus­trat­ing, and of­ten I feel as if I’ll nev­er do an­oth­er thing again, and I of­ten feel stu­pid be­cause there are al­ways people around me who seem to un­der­stand things faster than I do. Yet, when I learn something new it feels so good! Also, if I work with some­body else, and it’s a good piece of math­em­at­ics, we get to know each oth­er on a level that is very hard to come by in oth­er friend­ships. I learn things about how people think, and I find it very mov­ing and in­ter­est­ing. Math­em­at­ics puts me in touch with people on a deep level. It’s the cre­ativ­ity that oth­er people ex­press that touches me so much. I find that, and the math­em­at­ics, very beau­ti­ful. There is something very last­ing about it also.

No­tices: Let’s go back to the con­nec­tions between your work and com­plex­ity the­ory. Did you come up with an al­gorithm that can tell wheth­er a knot is the trivi­al knot?

Birman: Yes. But the al­gorithm that Hirsch and I dis­covered [6] is slow on simple ex­amples, and it is slow as the com­plex­ity of the ex­ample grows. Yet it has the po­ten­tial to be a poly­no­mi­al al­gorithm, and I don’t think that’s the case for the more fash­ion­able al­gorithms com­ing from nor­mal sur­face the­ory. There is a mis­un­der­stand­ing of our pa­per. Read­ers who did not read care­fully saw that we used nor­mal sur­faces in our pa­per (in a some­what tan­gen­tial man­ner). They dis­missed our pa­per as be­ing de­riv­at­ive, but it was not. There are ideas in our work that were ig­nored and not de­veloped.

However, at the present mo­ment it seems most likely that the prob­lem of al­gorith­mic­ally re­cog­niz­ing the un­knot will be solved via Hee­gaard Flo­er knot ho­mo­logy. That is a very beau­ti­ful new ap­proach, and for­tu­nately there is an army of gradu­ate stu­dents work­ing on it and mak­ing rap­id pro­gress. It was, some­how, fash­ion­able from day one and re­ceived lots of at­ten­tion. That can make a big dif­fer­ence in math­em­at­ics.

No­tices: Are there con­nec­tions between this and the P versus NP prob­lem?

Birman: Yes, there are con­nec­tions, but they are not dir­ectly re­lated to the un­knot al­gorithm. A prob­lem that has been shown to be NP-com­plete is “non-shortest words in the stand­ard gen­er­at­ors of the braid group”. If you had an al­gorithm to show that a word in the stand­ard gen­er­at­ors of the braid group is not the shortest rep­res­ent­at­ive of the ele­ment it defines, and could do that in poly­no­mi­al time, then you would have proved that P is equal to NP. Of course, if you are giv­en any word in the gen­er­at­ors of the braid group and want to know wheth­er it is shortest or not, all you have to do is try all the words that are short­er than it — and since there is a poly­no­mi­al solu­tion to the word prob­lem, you can test quickly wheth­er any fixed word that’s short­er than the one that you star­ted with rep­res­ents the same ele­ment. However, the col­lec­tion of all words that are short­er than the giv­en one is ex­po­nen­tial, so that solu­tion to the non­shortest word prob­lem is ex­po­nen­tial. But the nor­mal forms that I am work­ing on in the braid group are such that if you could un­der­stand them bet­ter, you might learn how to im­prove this test. But I am not hold­ing that up as a goal. At the mo­ment it seems like a ques­tion that is out of reach.

I have been work­ing on a re­lated ques­tion: the con­jugacy search prob­lem in the braid group. It’s com­plic­ated and dif­fi­cult, but I be­lieve strongly that it won’t be long be­fore someone proves that it has a solu­tion that’s poly­no­mi­al in both braid in­dex and word length. It’s a mat­ter of un­der­stand­ing the com­bin­at­or­ics well enough. It is re­lated to (but con­sid­er­ably weak­er than) the P versus NP prob­lem. I am work­ing on that prob­lem right now with two young math­em­aticians, Juan González-Me­neses from Seville, Spain, and Volk­er Gebhardt from Sydney, Aus­tralia.

No­tices: It’s amaz­ing that knot the­ory and braids are con­nec­ted to so many things.

Birman: I think I was very lucky be­cause my Ph.D. thes­is led me to many dif­fer­ent parts of math­em­at­ics. The par­tic­u­lar prob­lems that are sug­ges­ted by braids have led me to knot the­ory, to op­er­at­or al­geb­ras, to map­ping class groups, to sin­gu­lar­ity the­ory, to con­tact to­po­logy, to com­plex­ity the­ory and even to ODE [or­din­ary dif­fer­en­tial equa­tions] and chaos. I’m work­ing in a lot of dif­fer­ent fields, and in most cases the braid group had led me there and played a role, in some way.

No­tices: Why do braids have all these dif­fer­ent con­nec­tions?

Birman: Braid­ing and knot­ting are very fun­da­ment­al in nature, even if the con­nec­tions do not jump out at you. They can be subtle.

No­tices: Which res­ult of yours gave you par­tic­u­lar pleas­ure?

Birman: There are many ways to an­swer that ques­tion. I have had much pleas­ure from dis­cov­er­ing new math­em­at­ics. That happened, for ex­ample, when I was work­ing on my thes­is. The area was rich for the dis­cov­ery of new struc­ture, and (un­like most stu­dents) I ex­per­i­enced very little of the usu­al suf­fer­ing, to bring me down from that high. I have also got­ten much pleas­ure from col­lab­or­a­tions and the friend­ships they brought with them. I would prob­ably single out my good friend Bill Menasco as one of the best of my col­lab­or­at­ors. It has been a par­tic­u­lar pleas­ure to me when oth­ers have built on my ideas, and I see them grow in­to something that will be there forever, for oth­ers to en­joy. In that re­gard, I would single out the work that was done by Den­nis John­son in the 1980s, which built in part on the cal­cu­la­tion I had done alone in Par­is in 1971 and in an­oth­er part on my joint work with Robert Craggs [4]. In a re­lated way, I get great pleas­ure when I un­der­stand an idea that came from way back. An ex­ample was when I read sev­er­al pa­pers of J. Nielsen from the 1930s on map­ping class groups. (I had to cut open the pages in the lib­rary, they had been over­looked for a long time.) Nielsen’s great pa­tience and care in ex­plain­ing his ideas, and their ori­gin­al­ity and beauty, reached out over the years. I also feel priv­ileged to have worked as an ad­visor of very tal­en­ted young people and to have been a par­ti­cipant in the pro­cess by which they found their own cre­at­ive voices.

It would be dis­hon­est not to add that the com­pet­it­ive as­pect of math is something I dis­like. I also find that the pleas­ure in vari­ous hon­ors that have come to me is not so last­ing and have the dis­agree­able as­pect of mak­ing me feel un­deserving. The pleas­ure in ideas and in work well done is, on the oth­er hand, last­ing. But it’s easy to for­get that.

Women in mathematics

No­tices: The situ­ation for wo­men in math­em­at­ics has changed greatly. Have all the prob­lems been solved?

Birman: No, of course not. The dis­par­ity in the num­bers of men and wo­men at the most pres­ti­gi­ous uni­versit­ies (and I in­clude Columbia in that) is strik­ing. Any­one who enters a room in the math build­ing at Columbia when a sem­in­ar is in pro­gress can see it.

No­tices: Do you think at­ti­tudes to­ward wo­men in math­em­at­ics have im­proved?

Birman: Enorm­ously, in my life­time. On the whole, I think the pro­fes­sion is now very ac­cept­ing of wo­men. When I took my first job I was the first wo­man fac­ulty mem­ber at Stevens In­sti­tute of Tech­no­logy. A few years later, I was the only wo­man fac­ulty mem­ber (and I was a vis­it­or) in the Prin­ceton math de­part­ment. Now one sees ever-in­creas­ing num­bers of wo­men fac­ulty mem­bers, al­though the num­bers in the top re­search fac­ulties are still very small. That is cer­tainly the case at Columbia, but this year for the first time, Columbia’s fresh­man class of gradu­ate stu­dents was half men, half wo­men. Just six years ago it was all men, no wo­men.

Re­cently sev­er­al young people I know who are hus­band-and-wife math­em­aticians have got­ten jobs in the same de­part­ment. There used to be nepot­ism rules against that. It’s such a big ef­fort for a de­part­ment to make, to hire two people at the same time, in whatever fields they hap­pen to be in, some­times the same field. It’s im­press­ive that de­part­ments care enough about do­ing right by wo­men to do it. So yes, I think things are chan­ging.

But there are ser­i­ous is­sues re­gard­ing wo­men in re­search. At the mo­ment there are a very small num­ber of wo­men at the top of the pro­fes­sion. This is the very thing that Lawrence Sum­mers [former Har­vard Uni­versity pres­id­ent] poin­ted out. What are the reas­ons for it, and what can we do about it? It would be good to try to un­der­stand why, and if we don’t ad­mit all pos­sib­il­it­ies, then we may nev­er find out. So I was rather shocked that wo­men on the whole did not want to look at that prob­lem openly.

No­tices: He of­fen­ded a lot of wo­men when he spec­u­lated that there might be a bio­lo­gic­al dif­fer­ence between men and wo­men that ac­counts for the dif­fer­ence of per­form­ance.

Birman: Yes, he of­fen­ded, but the re­ac­tion “stop, don’t ask that ques­tion” was not a good re­sponse. Wo­men in math have done so much to help oth­er wo­men, and the is­sues are so com­plex, that I was dis­tressed that polit­ic­al cor­rect­ness over­shad­owed the need to un­der­stand things bet­ter. The truth may not al­ways be pleas­ant, but let’s find out what it is. If wo­men math­em­aticians re­fuse to face the is­sue openly, then who will do it for them? The so­ci­olo­gists? I hope not. However, that kind of dis­cus­sion is not my strong point. I am too opin­ion­ated and tact­less to say what needs to be said. Ral­ph Fox gave me tongue-in-cheek ad­vice long ago: “Speak of­ten and not to the point, and soon they will drop you from all the com­mit­tees.”

I did, however, won­der for many years wheth­er there was a way for me to help oth­er wo­men. Rather early in my ca­reer I began to work with male gradu­ate stu­dents, and I en­joyed that very much. Yet the first time a Columbia wo­man gradu­ate stu­dent ( Pei-Jun Xu, Ph.D. Columbia 1987) asked wheth­er she could work with me, my private re­ac­tion was “to­geth­er we will prob­ably make a total mess of it!”. We did not, and she wrote a fine thes­is, and on the way I un­der­stood that I could help her in more ways than math just be­cause we were both wo­men and I sensed some of her un­spoken con­cerns. Ever since then I real­ized that was the unique way that I could help oth­er wo­men — simply by tak­ing an in­terest, work­ing with them when it was ap­pro­pri­ate, and be­ing open to their con­flicts and sens­it­ive to their con­cerns.

No­tices: That’s what it comes down to, the wo­men ac­tu­ally do­ing math­em­at­ics.

Birman: Yes, of course it does. I have heard some wo­men who are bit­ter be­cause they feel the re­wards of re­search don’t seem big enough for the sac­ri­fice. Of course there are men who feel that way too. Fritz John, a very fine re­search math­em­atician, once said to me that at the end of the day the re­ward was “the grudging ad­mir­a­tion of a few col­leagues”. Well, if what you are look­ing for is ad­mir­a­tion be­cause you have done a great piece of work, ad­mir­a­tion is of­ten not there (and maybe the work isn’t so great either). What is much more im­port­ant, to me, is when some­body has really read and un­der­stood what I have done, and moved on to do the next thing. I am thrilled by that. Sure, it’s nice to get a gen­er­ous ac­know­ledg­ment, but that is a bo­nus. The real pleas­ure is to be found in the math­em­at­ics.


[1] J. S. Birman: “On Siegel’s mod­u­lar group,” Math. Ann. 191 (1971), pp. 59–​68. MR 0280606 Zbl 0208.​10601 article

[2] J. S. Birman and H. M. Hilden: “On the map­ping class groups of closed sur­faces as cov­er­ing spaces,” pp. 81–​115 in Ad­vances in the the­ory of Riemann sur­faces (Stony Brook, NY, 1969). Edi­ted by L. V. Ahlfors, L. Bers, H. M. Far­kas, R. C. Gun­ning, I. Kra, and H. E. Rauch. An­nals of Math­em­at­ics Stud­ies 66. Prin­ceton Uni­versity Press, 1971. MR 0292082 Zbl 0217.​48602 incollection

[3] J. S. Birman: Braids, links, and map­ping class groups. An­nals of Math­em­at­ics Stud­ies 82. Prin­ceton Uni­versity Press, 1974. Based on lec­ture notes by James Can­non. An er­rat­um to The­or­em 2.7 is giv­en in Can. J. Math. 34:6 (1982). MR 0375281 Zbl 0305.​57013 book

[4] 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

[5] 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

[6] J. S. Birman and M. D. Hirsch: “A new al­gorithm for re­cog­niz­ing the un­knot,” Geom. To­pol. 2 (1998), pp. 175–​220. MR 1658024 Zbl 0955.​57005 article

[7] J. S. Birman and W. W. Menasco: “Sta­bil­iz­a­tion in the braid groups, I: MT­WS,” Geom. To­pol. 10 : 1 (2006), pp. 413–​540. MR 2224463 Zbl 1128.​57003 article

[8] J. S. Birman and W. W. Menasco: “Sta­bil­iz­a­tion in the braid groups, II: Trans­vers­al sim­pli­city of knots,” Geom. To­pol. 10 : 3 (2006), pp. 1425–​1452. MR 2255503 Zbl 1130.​57005 ArXiv math.​GT/​0310280 article