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 ofWilhelm 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 andAlgebraic 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 byNotices Deputy Editor Allyn Jackson and Associate Editor Lisa Traynor.
Notices:Let’s start at the beginning of your life. Wereyour 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, tosearch for lost relatives and to seek a better life. My mother wasborn in New York, but her parents were immigrants fromRussia–Poland.
Notices:What did your father do?
Birman: He started as a shipping clerk in the dress industry andworked his way up to become a successful dress manufacturer. He toldhis four daughters repeatedly that the U.S. was the best country inthe 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 highschool.
Notices:Why did they emphasize their four daughters getting an education?
Birman: Jewish culture, as it was handed down to us, included thestrong belief that Jews survived for so many years in the Diasporabecause they were “the people of the book”. The free translation, whenI brought home an exam with a grade of 98, was “what happened to theother 2 points?” Becoming an educated person, and using that educationto do something bigger than just to earn money, was set up to mygeneration 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 earlierthan 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 certainlyhad challenging math. I went to an all-girls high school in NewYork, Julia Richmond High School. It was really a rough inner-cityhigh school, but within it there was a small academic unit, a schoolwithin a school. We had some very good teachers. We had a course inEuclidean geometry, and every single night we would have telephoneconversations and argue over the solutions to the geometry problems.That was my introduction to proof, and I just loved it, it waswonderful. When the course ended, I joined a small group of girls whocampaigned for more geometry, but the teacher (her name was MissMahoney) was willing but perhaps not knowledgeable enough to know howto continue to challenge the intellectual interests of this eagergroup of girls! She taught us 3-dimensional Euclidean geometry, andthat was a little dull. If she had taught us hyperbolic geometry, orgroup theory, where we would have encountered new ideas, we wouldhave been in heaven!
Notices:Usually high school girls are on thephone talking about their hair.
Birman: We did that too! Actually Iwas in this little group, and we were definitely regarded as beingnerds. Most of the girls in our selective school within a schoolworked hard and got good grades, but talked all the time about boysand clothes. I was a late developer and wasn’t ready for that. Ididn’t date at all until I was in college. Still, at one point I waselected president of the class, so the other students could not havebeen really hostile. I felt accepted, and even liked. There was anatmosphere 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 becamea plant physiologist. She was Ruth Satter of the Satter Prize. Shehad a fine academic career, before her untimely death from leukemia.Helen is independently wealthy and is a philanthropist, with veryspecial interests of her own. My younger sister, Ada, became akindergarten 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 wasadvised to take at Swarthmore was a cookbook calculus course, and itwas both boring and unconvincing. So I looked around and found otherthings 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 mathofferings were all low-level. When you got to the point where you wereready for serious math, you were directed to courses at Columbia,which at that time was an all-male school. That was the first timethat I hit a situation where I was one of a very small number ofgirls. 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 ideathat maybe math was not for girls.
Birman: Yes. But there was a long gap before I went on to graduateschool. 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 nonsenselike that. But on the other hand, the only way that a respectable girlcould get out from under her parents control was to marry, so I wasnot averse to the idea. But I did not want to make a mistake in mychoice, and that took attention. I did think about going to graduateschool, but I understood how hard math was. I thought it would takelots of concentrated effort, as it must for any serious student. I wasafraid that I would wreck my life if I gave math that kind ofattention at that time. (I think I was right. As we talk, Joe and Ihave been married for fifty-six years, and he has been my biggestsupporter.) Actually, I didn’t really decide not to go to graduateschool, but when the opportunity arose to put it off and accept aninteresting job, the job was appealing.
The job was very nice. I was extremely lucky. It was at an engineeringfirm that made microwave frequency meters. These meters werecylindrical cans with two parameters, the radius of the base and thedepth (or height). The radius was fixed, but the depth could bechanged with a plunger, changing the resonant frequency. The(depth-to-resonant frequency) curve was nonlinear, and the problemwas that they had a hard time calibrating the dials, putting thenotches on to indicate what the frequency would be as you pushed theplunger in. They hired me because they had the idea that they couldsell more meters if they could push in the plunger in a novel way thatwould yield an approximately linear response curve. In calculus Ihad learned about ladders sliding against a wall, and in the jobinterview the idea came up that the curve that gave the height of theladder as a function of its distance from the wall might be a curvethat could be fitted to the experimental data. The idea worked verywell. For about eight months I computed the parameters, and theyconstructed meters of all sizes with plungers that pushed in along anaxis orthogonal to the axis of the can. The dials were for allpractical purposes linear. I was very happy!
But when that project ended, they set me to work taking measurementson an oscilloscope, and that was pretty dull. One day I happened torun into my old physics professor from Barnard, and he offered me aposition as the physics lab assistant at Barnard. I took the positionand applied to graduate school in physics. I realized that my jobpossibilities would improve if I had a physics degree. I did get amaster’s degree in physics, but I do not have good intuition for thesubject. I felt they could just tell me anything, and I would have tobelieve it. I am astonished these days at the way in which physicshas fed into math. Physicists do seem to have an intuition that goesbeyond what mathematicians very often see, and they have differenttests of truth. I just didn’t have that intuition. Yet I reallyenjoyed the physics lab, because when I saw things in the lab, I knewthey were true. But I didn’t always trust the laws of physics that welearned. 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 anavigation computer. The pilot would be flying a plane, and thecomputer would send a radar signal to the ground. The signal wouldbe bounced back to the plane. The computer measured the Dopplershift and used it to compute air speed and altitude. My part of thewhole thing was error analysis — to figure out the errors when the planewas being bounced around by changes in air pressure. A second problemwas that of maximizing aircraft range for a fixed amount of fuel. Athird was the design of a collision avoidance system.
Notices:Were there many women?
Birman: I worked at three different engineering firms. At one of themthere were several women, but at the others I don’t recall any otherwomen.
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 yearslater. When my first child was born, I planned to go back to workbecause I really liked what I was doing. But that posed a problem. Inthose days, there was no day care. Unless you had a family member totake care of your children (and my mother and mother-in-law wereunable to do that), it was almost impossible. My husband and I hadthought, very unrealistically, we will put an ad in the paper and hiresomebody. But then, I had this huge responsibility for our baby, and Ijust couldn’t see leaving him with somebody about whom I knew verylittle. 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 oneday a week after we had a second child. Just before our third childwas born, my husband had been invited to teach in a distant city. Hehad 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 alreadydwindled down to one day a week. When I came back I knew I couldn’twork that way anymore. So I went to graduate school with the idea oflearning some new things for when I’d go back to work. You can seethat I led a very wandering and undirected life! It amazes me that Igot 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 wasborn, on January 12, 1961. I went to New York University, where myhusband was on the faculty, so that my tuition was free. NYU’s CourantInstitute had an excellent part-time program, with evening coursesthat were essentially open admissions. I took linear algebra the firstsemester, 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 LouisNirenberg. In the first lecture he said, “A complex number is a pairof real numbers, with the following rules for adding and multiplying them.”I certainly knew about “imaginary numbers”, but he putthem into a framework that was sound mathematics. It sounds like atrivial change, but it was not. Eventually, I also had a course intopology, which I loved, withJack Schwartz. He was not a topologist,and when I go back and look at my notes, I see it was a weird topologycourse! He was somebody who liked to try new things. He taught uscohomology in a beginning topology course — not homology, not even thefundamental group! But I really loved that course. It really grabbedme, although the approach had its down side, as I knew almost noexamples. I had started studying at Courant with the intention oflearning some applied mathematics. But everything I learned pushedme toward pure mathematics.
At Courant I was starting to pile up enough courses for an MA, andthere was a required master’s final exam. When I took the exam, Ididn’t realize it was also the Ph.D. qualifying exam. I wassurprised when I passed it for the Ph.D. That’s when I applied forfinancial 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 manywomen around. The people in the department were very nice to me — theyrealized that I had three children, and they did not give me heavy TAassignments.Karen Uhlenbeck was one of the students there, but shetransferred out.Cathleen Morawetz was on the faculty, and I took onecourse 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 seriesof more specialized exams for admission to research. My husband was onthe NYU faculty, and the first question I was asked in one of theexams was, “Who is smarter, you or your husband?”
Notices:That was the first question?
Birman: Yes, it’s ludicrous, in 2006. Later onwhen I became a mathematician, I met the person who asked thisquestion and reminded him of it, and he said, “Oh no, not me! I didn’tsay that!”
Notices:How did you answer the question?
Birman: I laughed. It was the only thing to do. Afterwards I started to getreally angry about it. It was a stupid question!
Anyway, I passed that exam too and went looking for an advisor. Thefirst person I approached was the topologistMichel Kervaire, but hewasn’t interested. He said, “You’re too old and you don’t know enoughtopology.” He was right, I didn’t know enough topology. And I canunderstand why he would be skeptical of a person my age. You haveto be convinced when you see someone who is outside of the usualframework that the person is a serious student, and he had never beenmy classroom teacher.
I went to speak toNirenberg. He was very helpful to me. I read theNotices interview with him, and he had told you that he lovedinequalities. That’s funny, because I remember he asked me, “Do youlike inequalities?” And I said, “No, I don’t like inequalities!” Hesaid, “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 toWilhelm Magnus. He had noticed me, because I had done some grading forhim. 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. Helater told me of his habit of picking up strays, and in some way I wasa stray.
Notices:What paper was it that he gave you?
Birman: It was a paper byFadell andNeuwirth[e1].
The braid groupswere defined in that paper as the fundamental group of a certainconfiguration space. Magnus said that he didn’t understand thedefinition, and it took me a long time to understand it. Finally Idid, and I was very happy. Magnus had worked on the mapping classgroup of a twice-punctured torus, and he had suggested that I couldextend this work to a torus with 3 or 4 punctures. My thesis ended upbeing about the mapping class group of surfaces of any genus with anynumber of punctures. He thought that was a real achievement. As soonas I understood the problem well enough, I solved it. It was both funand very encouraging.
Around this time there was a very differentpaper byGarside on braids that interested me greatly[e2].I was awareof the fact that there was a scheme for classifying knots with braids.When I saw that Garside had solved the conjugacy problem in thebraid group, I thought that was going to solve the knot problem. Icouldn’t have been more mistaken, but still, it grabbed my interest. Iam still working on it — right now I am trying to show that Garside’salgorithm can be made into a polynomial algorithm. This is importantin complexity theory. So my interest in that problem dates back tograduate school.
Notices:After you got your Ph.D., you got a job at Stevens Instituteof Technology.
Birman: I had not done a thorough job on applications and was notoffered any job until late August 1968, when Stevens Institute hadsome unexpected departures. The first year I was there I startedworking withHugh M. Hilden (who is known as Mike). We solved a neatproblem that year and wrote several really good papers. The one I likebest is the first in the series[2].
The work with Hilden was very rewarding. My thesis had been on themapping class group of a punctured surface. I showed there is ahomomorphism from the mapping class group of a punctured surface tothat of a closed surface, induced by filling in the punctures. Iworked out the exact sequence that identified the kernel of thathomomorphism, but I didn’t know a presentation for the cokernel, themapping class group of a closed surface, and realized that was aproblem that I would like to solve. The whole year I talked about itto Mike, whose office was next to mine, and finally we solved theproblem for the special case of genus 2. As it turned out, oursolution had many generalizations, but the key case was a closedsurface\( \Sigma \) of genus 2. In that case, the mapping class group has acenter, and the center is generated by the class of an involution thatI’ll call\( \mathcal{I} \). The orbit space\( \Sigma/\mathcal{I} \) is a2-sphere\( S^2 \), and the orbitspace projection\( \Sigma \rightarrow \Sigma/\mathcal{I}= S^2 \) gives it thestructure of a branchedcovering space, the branch points being the images on\( S^2 \) of the 6fixed points of\( \mathcal{I} \). We were able touse the fact that the mapping class group of\( S^2_6 \) of2 minus those 6 points was a known group (related to the braid group),to find a presentation forthe mapping class group\( \mathcal{M}(\Sigma) \) of\( \Sigma \). The difficultywe had to overcome was that mapping classes arewell-defined only up to isotopy. We knew that ingenus 2, every mapping class was represented bya map that commuted with\( \mathcal{I} \), but we did not knowwhether every isotopy could be deformed to a newisotopy that commuted with\( \mathcal{I} \). We felt it had to betrue, but we couldn’t see how to prove it. One dayMike and I had the key idea, together. The idea wasto look at the path traversed on\( \Sigma \) by one of the 6fixed points, say\( p \), under the given isotopy. Thispath is a closed curve on\( \Sigma \) based at\( p \). Could thatclosed curve represent a nontrivial element in\( \pi_1(\Sigma,p) \)? It was a key question. Once we asked theright question, it was easy to prove that the answerwas no, and as a consequence our given isotopycould be deformed to one that projected to an isotopy on\( S^2_6 \). As aconsequence, there is a homomorphism\( \mathcal{M}(\Sigma) \rightarrow\mathcal{M}(S^2_6) \), withkernel\( \mathcal{I} \). Ourhoped-for presentation followed immediately. Itwas a very fine experience to work with Mike, toget to know him as a person via shared mathematics. It was the first timeI had done joint work,and I enjoyed it so much that ever since I have beenalert to new collaborations. They are different eachtime, but have almost all been rewarding.
At that point I was thoroughly involved in mathematics. But myhusband had a sabbatical, and I had promised him that I would take ayear off so that he could spend his sabbatical with collaborators inFrance. So I took a leave of absence from my job and found myself inParis, and in principle it should have been a lovely year. But we hadthree children, and once again I had lots of home responsibilities!Moreover, I didn’t know any of the French mathematicians, because Ihad come to France without any real introductions, and nobody wasinterested in braids. French mathematics at that time was heavilyinfluenced by theBourbaki school. I found myself very isolated anddiscouraged. Looking for a problem that I could handle alone, I decided todo a calculation.
There is a homomorphism from the mapping class group of a surface tothe symplectic group. People knew defining relations for thesymplectic group, but not for the mapping class group, unless thegenus is\( \leq 2 \). I was interested in the kernel of that homomorphism,which is called the Torelli group. It was an immense calculation. Ifinished it, and I did get an answer[1],which was later improvedwith the help of a Columbia graduate student,Jerome Powell. In 2006a graduate student at the University of Chicago,Andy Putman, constructedthe first conceptual proof of the theorem that Powell and Ihad proved. Putman’s proof finally verifies the calculation I did thatyear in France!
When I returned from France I was invited to give a talk at Princetonon the work that Hilden and I had done together. That was when mycareer really began to get going, because people were interestedin what we had done. I was invited to visit Princeton the followingyear. 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 interestedgroup, includingRalph Fox andKunio Murasugi, andJames Cannon, atthat 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 knottheory had been largely undeveloped. Three topics that I developedin the lectures and put into the book were: (1) Alexander’s theoremthat every link type could be represented, nonuniquely, by a closedbraid, (2) Markov’s theorem, which described the precise way inwhich two distinct braid representatives of the same link type wererelated, one of those moves being conjugacy in the braid group, and(3) Garside’s solution to the problem of deciding whether twodifferent braids belonged to the same conjugacy class. I had chosenthose topics because I was interested in studying knots via closedbraids, 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 knownproof of Markov’s theorem! Markov had announcedit in 1935, and he had sketched a proof but did notgive details, and the devil is always in the details.When I told my former thesis advisor, WilhelmMagnus, he remarked that the sketched proof wasvery likely wrong! But luckily, I was able to followMarkov’s sketch, with the help of some notes thatRalph Fox had taken at a seminar lecture given bya former Princeton grad student (his name vanishedwhen he dropped out of grad school). After somenumber of 2:00 a.m. bedtimes I was able to presenta proof. There are now some six or seven conceptually different proofs ofthis theorem, but the onein my 1974 book was the first.
Notices:Can you tell us about your interaction withVaughan Jones,when he was getting his ideas about his knot polynomial?
Birman: One day in early May 1984, Vaughan Jones called to askwhether we could get together to talk about mathematics. He contactedme because he had discovered certain representations of the braidgroup and what he called a “very special” trace function on them, andpeople had told him that I was the braid expert and might have someideas 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 workedin very different parts of mathematics and we had the expecteddifficulties in understanding each other’s languages. His trace arosein his work on von Neumann algebras, and it was related to the indexof a type\( \mathrm{II}_1 \) subfactor in a factor. All that was far away from braidsand links. When we met, I told him about Alexander’s theorem, andMarkov’s theorem, and Garside’s work. He told me about his representationsand about his trace function. Of course, his explanationswere given in the context of operator algebras. I recall that I saidto him at one point, Is your trace a matrix trace? And he said no, itwas not. Well, that answer was correct, but he did not say that histrace was a weighted sum of matrix traces, and so I did not realizethat, if one fixed the braid index, the trace was a class invariantin the braid group. He understood that very well and did notunderstand what I had missed. He would willingly have said more, if hehad, because he is super-generous and truly decent. In between ourmeetings he gave the matter much thought (which I did not!), and onenight he had the key idea that by a simple rescaling of his trace, itwould in fact become invariant under all the moves of Markov’stheorem, and so become a link invariant. He told me all this, in greatexcitement, on the telephone. The proof that his normalized trace wasa link invariant was immediate and crystal clear. After all, a goodpart of my book had been written with the goal of making the Alexanderand Markov theorems into useful tools in knot theory, and Vaughan hadused them in a very straightforward way.
Was his new invariant really new, or a new way to look at somethingknown? He did not know. Examples were needed, and a few days laterwe met again, in my office, to work some out. That was probably May22, 1984. The new link invariant was a Laurent polynomial. My firstthought was: it must be the Alexander polynomial. So I said, “Here aretwo knots (the trefoil and its mirror image) that have the sameAlexander polynomial. Let’s see if your polynomial can distinguishthem.” To my astonishment, it did! Well, we checked that calculationvery carefully, on lots more examples, because the implications werehard to believe. By pure accident, I had recently worked out aclosed braid representative of the Kinoshita–Terasake 11-crossingknot, whose Alexander polynomial was zero. Fishing it out of my filecabinet we learned very quickly, that same day, that the newpolynomial was nonzero on it. So in just that one afternoon, we knewthat 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 thinkingthat nobody else knows this thing exists! It was an amazing discovery.Very quickly, other parts of the new machinery came to bear, and theworld of knot theory experienced an earthquake. There was not just theJones polynomial, but also its cousins, the HOMFLY and the Kaufmanpolynomials, and lots more. And some of the stuff in my book aboutmapping class groups was relevant too. Much later, Garside’s machineryappeared too, in a particular irreducible representation of the braidgroup that arose via the same circle of ideas. Garside’s solution tothe word problem was used byDaan Krammer to prove that braid groupsare linear.
There was another related part to this story. In 1991Vladimir Arnoldcame to the United States to visit Columbia for a semester. I knewArnold and met him in the lobby as he arrived, in September, with hissuitcase. He is a very excitable and enthusiastic man. He put downhis suitcase right then and there and opened it on the floor next tothe elevator to get out a paper he had brought for me. It was by hisformer studentViktor Vassiliev. He said, “You have to read thispaper, it’s wonderful, it contains new knot invariants, and they comefrom singularity theory, and it’s fine work, and I would like yourhelp in publicizing it!” Of course I looked at the paper. At thatpoint there had been an explosion in new knot invariants, and the openquestion was what they meant geometrically. And here Arnold was, withmore invariants! The old ones were polynomials, the new ones wereintegers (lots of integers!). Arnold asked me to copy and distributethe paper in the United States. So one afternoon shortly after hisarrival I made lots of photocopies, and sent them out to everyone Icould think of who seemed appropriate. But even as I did it Isuspected the knot theory community might not be so overjoyed to haveyet more knot invariants coming unexpectedly out of left field! Thereis resistance to learning new things. We had just learned aboutoperator algebras, and suddenly we had to learn about singularitytheory! But Arnold kept after me, at tea every day.
Xiao-Song Lin was an assistant professor in the department, and hisfield 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. Itold him about the paper of Vassiliev. We read it together, and wefinally understood most of it. We said, here are the Vassiliev invariants,and there are the knot polynomials — and they must be related insome way. But how? For a fixed knot or link, its Jones polynomial wasa one-variable Laurent polynomial with integer coefficients, whereasits Vassiliev invariants were an infinite sequence of integers, orpossibly of rational numbers.
We had an idea that perhaps we should, for the moment, set aside thefact that the Vassiliev invariants came from the machinery ofsingularity theory, and try to construct them from their properties.We did that because we knew that the Jones polynomial (the simplest ofthe knot polynomials) could be constructed from its properties. Wethought that might be a way for us see a connection. That had goodand bad consequences. The bad one was that later, Vassiliev invariantswere renamed “finite type invariants”, and were defined via ouraxioms. In the process their origins in singularity theory were lostand remain underdeveloped to this day.
Soon Lin and I realized howto make the connection we had been seeking. We had the idea ofmaking a change of variables in the Jones polynomial, changing itsvariable from\( x \) to\( t \), with\( x = e^t \).
The Jones polynomial was a Laurent polynomial in\( x \), and\( e^{kt} \) has anexpansion in positive powers of\( t \) for every positive and negativeinteger\( k \). This change in variables changes the Jones polynomial toan infinite series in powers of\( t \). We were able to prove that thecoefficients in that infinite series satisfied all of our axioms forVassiliev invariants, and so were Vassiliev invariants[5].Everythingwent quickly with that idea — eventually all the knot polynomials wererelated to Vassiliev invariants in this way. They are generatingfunctions for particular infinite sequences of FT invariants. But infact the set of FT invariants is larger than those coming from knotpolynomials. They are more fundamental objects.
Notices:Can you tell us about your recent work withMenascothat involved the Markov theorem?
Birman: That is another aspect of the same underlying project, tounderstand knots through braids. In 1990 at the International Congressin Kyoto, when Vaughan Jones got the Fields Medal, I gave a talk onhis work. Afterward Bill Menasco invited me to give a colloquium basedon it in the math department in Buffalo. So I gave a talk there aboutVaughan Jones’s work, and I stayed at Bill’s house that night. Westarted to talk, and he said, “What problems are you working on?What’s your dream?” I told him my dream is to classify knots bybraids. I had an idea about how you could avoid the “stabilization”move in Markov’s theorem. Then about three weeks later, I got a letterfrom him saying “I have an idea how we might try to prove the‘Markov theorem without stabilization’ (MTWS).” And that’s when ourcollaboration began. Of course, my original conjecture was much toosimple. We kept solving little pieces of the sought-for theorem. Wewrote 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. BillMenasco and I are very good friends. We have had such a longcollaboration. But we have very different styles. He can sit in achair and stare at the ceiling as he works on mathematics, but Ilike 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’thave any obligations, and I keep right on working on math. Sometimesmathematics can be frustrating, and often I feel as if I’ll never doanother thing again, and I often feel stupid because there arealways people around me who seem to understand things faster than Ido. Yet, when I learn something new it feels so good! Also, if I workwith somebody else, and it’s a good piece of mathematics, we get toknow each other on a level that is very hard to come by in otherfriendships. I learn things about how people think, and I find it verymoving and interesting. Mathematics puts me in touch with people on adeep level. It’s the creativity that other people express that touchesme so much. I find that, and the mathematics, very beautiful. There issomething very lasting about it also.
Notices:Let’s go back to the connections between your work andcomplexity theory. Did you come up with an algorithm that can tellwhether a knot is the trivial knot?
Birman: Yes. But the algorithm thatHirsch and I discovered[6]isslow on simple examples, and it is slow as the complexity of theexample grows. Yet it has the potential to be a polynomial algorithm,and I don’t think that’s the case for the more fashionablealgorithms coming from normal surface theory. There is amisunderstanding of our paper. Readers who did not read carefully sawthat we used normal surfaces in our paper (in a somewhat tangentialmanner). 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 problemof algorithmically recognizing the unknot will be solved viaHeegaard Floer knot homology. That is a very beautiful new approach,and fortunately there is an army of graduate students working on itand making rapid progress. It was, somehow, fashionable from day oneand received lots of attention. That can make a big difference inmathematics.
Notices:Are there connections between this and the P versus NPproblem?
Birman: Yes, there are connections, but they are not directly relatedto the unknot algorithm. A problem that has been shown to beNP-complete is “non-shortest words in the standard generators of thebraid group”. If you had an algorithm to show that a word in thestandard generators of the braid group is not the shortestrepresentative of the element it defines, and could do that inpolynomial time, then you would have proved that P is equal to NP. Ofcourse, if you are given any word in the generators of the braid groupand want to know whether it is shortest or not, all you have to do istry all the words that are shorter than it — and since there is apolynomial solution to the word problem, you can test quicklywhether any fixed word that’s shorter than the one that you startedwith represents the same element. However, the collection of allwords that are shorter than the given one is exponential, so thatsolution to the nonshortest word problem is exponential. But thenormal forms that I am working on in the braid group are such thatif you could understand them better, you might learn how to improvethis test. But I am not holding that up as a goal. At the moment itseems like a question that is out of reach.
I have been working on a related question: the conjugacy searchproblem in the braid group. It’s complicated and difficult, but Ibelieve strongly that it won’t be long before someone proves that ithas a solution that’s polynomial in both braid index and word length.It’s a matter of understanding the combinatorics well enough. It isrelated to (but considerably weaker than) the P versus NP problem. Iam working on that problem right now with two young mathematicians,Juan González-Meneses from Seville, Spain, andVolker Gebhardt fromSydney, Australia.
Notices:It’s amazing that knot theory and braids are connected to somany things.
Birman: I think I was very lucky because my Ph.D. thesis led me tomany different parts of mathematics. The particular problems that aresuggested by braids have led me to knot theory, to operatoralgebras, to mapping class groups, to singularity theory, to contacttopology, to complexity theory and even to ODE [ordinary differentialequations] and chaos. I’m working in a lot of different fields, andin most cases the braid group had led me there and played a role, insome way.
Notices:Why do braids have all these differentconnections?
Birman: Braiding and knotting are very fundamental in nature, evenif 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 muchpleasure from discovering new mathematics. That happened, for example,when I was working on my thesis. The area was rich for thediscovery of new structure, and (unlike most students) I experiencedvery little of the usual suffering, to bring me down from that high. Ihave also gotten much pleasure from collaborations and thefriendships they brought with them. I would probably single out mygood friend Bill Menasco as one of the best of my collaborators. Ithas been a particular pleasure to me when others have built on myideas, 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 thatwas done byDennis Johnson in the 1980s, which built in part on thecalculation I had done alone in Paris in 1971 and in another part onmy joint work withRobert Craggs[4].In a related way, I get greatpleasure when I understand an idea that came from way back. An examplewas when I read several papers ofJ. Nielsen from the 1930s on mappingclass groups. (I had to cut open the pages in the library, they hadbeen overlooked for a long time.) Nielsen’s great patience and care inexplaining his ideas, and their originality and beauty, reached outover the years. I also feel privileged to have worked as an advisor ofvery talented young people and to have been a participant in theprocess by which they found their own creative voices.
It would be dishonest not to add that the competitive aspect of mathis something I dislike. I also find that the pleasure in varioushonors that have come to me is not so lasting and have the disagreeableaspect of making me feel undeserving. The pleasure in ideasand in work well done is, on the other hand, lasting. But it’s easy toforget that.
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 andwomen at the most prestigious universities (and I include Columbia inthat) is striking. Anyone who enters a room in the math building atColumbia 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 theprofession is now very accepting of women. When I took my first job Iwas the first woman faculty member at Stevens Institute of Technology.A few years later, I was the only woman faculty member (and I was avisitor) in the Princeton math department. Now one sees ever-increasingnumbers of women faculty members, although the numbers inthe top research faculties are still very small. That is certainly thecase at Columbia, but this year for the first time, Columbia’sfreshman class of graduate students was half men, half women. Just sixyears ago it was all men, no women.
Recently several young people I know who are husband-and-wifemathematicians have gotten jobs in the same department. There used tobe nepotism rules against that. It’s such a big effort for a departmentto make, to hire two people at the same time, in whateverfields they happen to be in, sometimes the same field. It’simpressive that departments care enough about doing right by womento do it. So yes, I think things are changing.
But there are serious issues regarding women in research. At themoment there are a very small number of women at the top of theprofession. This is the very thing that Lawrence Summers [formerHarvard University president] pointed out. What are the reasons forit, and what can we do about it? It would be good to try to understandwhy, and if we don’t admit all possibilities, then we may never findout. So I was rather shocked that women on the whole did not want tolook at that problem openly.
Notices:He offended a lot of women when he speculated that theremight be a biological difference between men and women that accountsfor the difference of performance.
Birman: Yes, he offended, but the reaction “stop, don’t ask thatquestion” was not a good response. Women in math have done so much tohelp other women, and the issues are so complex, that I was distressedthat political correctness overshadowed the need to understand thingsbetter. The truth may not always be pleasant, but let’s find out whatit is. If women mathematicians refuse to face the issue openly, thenwho will do it for them? The sociologists? I hope not. However, thatkind of discussion is not my strong point. I am too opinionatedand tactless to say what needs to be said. Ralph Fox gave metongue-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 meto help other women. Rather early in my career I began to work withmale graduate students, and I enjoyed that very much. Yet the firsttime a Columbia woman graduate student (Pei-Jun Xu, Ph.D. Columbia1987) asked whether she could work with me, my private reaction was“together we will probably make a total mess of it!”. We did not, andshe wrote a fine thesis, and on the way I understood that I could helpher in more ways than math just because we were both women and Isensed some of her unspoken concerns. Ever since then I realized thatwas the unique way that I could help other women — simply by taking aninterest, working with them when it was appropriate, and beingopen to their conflicts and sensitive to their concerns.
Notices:That’s what it comes down to, the women actually doingmathematics.
Birman: Yes, of course it does. I have heard some women who arebitter because they feel the rewards of research don’t seem big enoughfor the sacrifice. Of course there are men who feel that way too.Fritz John, a very fine research mathematician, once said to me thatat the end of the day the reward was “the grudging admiration of afew colleagues”. Well, if what you are looking for is admirationbecause you have done a great piece of work, admiration is often notthere (and maybe the work isn’t so great either). What is much moreimportant, to me, is when somebody has really read and understood whatI 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 abonus. The real pleasure is to be found in the mathematics.
[e1]E. Fadell and L. Neuwirth:“Configuration spaces,”Math. Scand.10(1962), pp.111–118.MR0141126Zbl0136.44104article
[e2]F. A. Garside:“The braid group and other groups,”Quart. J. Math. Oxford Ser. (2)20(1969), pp.235–254.MR0248801Zbl0194.03303article
[1]J. S. Birman:“On Siegel’s modular group,”Math. Ann.191(1971), pp.59–68.MR0280606Zbl0208.10601article
Siegel’s modular group\( \Gamma_g \) is the group of all\( 2g \times 2g \) matrices with integral entries which satisfy the condition:\[ SJS^{\prime} = J \] where\( s \in \Gamma_g \),\( S^{\prime} = \) transpose of\( S \), and if\( I_g \) and\( 0_g \) are the\( g\times g \) unit and zero matrices respectively, then:\[ J = \left( \begin{array}{cc} 0_g & I_g \\ -I_g & 0_g \end{array} \right). \] Generators for\( \Gamma_g \) were first determined by Hua and Reiner [1949], and Klingen [1961] obtained a characterization of\( \Gamma_g \) for\( g\geq 2 \) by a finite system of defining relations. However, Klingen’s results have been of somewhat limited use because, while he gives a procedure for finding defining relations, he does not carry it through explicitly, and in fact the explicit determination of such a system involves a fairly tedious and lengthy calculation. Finding ourselves in the position of needing explicit information about\( \Gamma_g \), we set ourselves the task of reducing Klingen’s results to more useable form. The primary purpose of the paper is to give the results of this calculation (Theorem 1) and to describe the methodology behind our proof.
@article {key0280606m,AUTHOR = {Birman, Joan S.},TITLE = {On {S}iegel's modular group},JOURNAL = {Math. Ann.},FJOURNAL = {Mathematische Annalen},VOLUME = {191},YEAR = {1971},PAGES = {59--68},DOI = {10.1007/BF01433472},NOTE = {MR:0280606. Zbl:0208.10601.},ISSN = {0025-5831},}[2]J. S. Birman and H. M. Hilden:“On the mapping class groups of closed surfaces as covering spaces,” pp.81–115inAdvances in the theory of Riemann surfaces(Stony Brook, NY, 1969). Edited byL. V. Ahlfors, L. Bers, H. M. Farkas, R. C. Gunning, I. Kra, and H. E. Rauch.Annals of Mathematics Studies66.Princeton University Press,1971.MR0292082Zbl0217.48602incollection
@incollection {key0292082m,AUTHOR = {Birman, Joan S. and Hilden, Hugh M.},TITLE = {On the mapping class groups of closed surfaces as covering spaces},BOOKTITLE = {Advances in the theory of {R}iemann surfaces},EDITOR = {Ahlfors, Lars V. and Bers, Lipman and Farkas, Hershel M. and Gunning, Robert C. and Kra, Irwin and Rauch, Harry E.},SERIES = {Annals of Mathematics Studies},NUMBER = {66},PUBLISHER = {Princeton University Press},YEAR = {1971},PAGES = {81--115},NOTE = {(Stony Brook, NY, 1969). MR:0292082. Zbl:0217.48602.},ISSN = {0066-2313},ISBN = {9780691080819},}[3]J. S. Birman:Braids, links, and mapping class groups.Annals of Mathematics Studies82.Princeton University Press,1974.Based on lecture notes by James Cannon.An erratum to Theorem 2.7 is given inCan. J. Math.34:6 (1982).MR0375281Zbl0305.57013book
| James Weldon Cannon | Related |
@book {key0375281m,AUTHOR = {Birman, Joan S.},TITLE = {Braids, links, and mapping class groups},SERIES = {Annals of Mathematics Studies},NUMBER = {82},PUBLISHER = {Princeton University Press},YEAR = {1974},PAGES = {ix+228},NOTE = {Based on lecture notes by James Cannon. An erratum to Theorem 2.7 is given in \textit{Can. J. Math.} \textbf{34}:6 (1982). MR:0375281. Zbl:0305.57013.},ISSN = {0066-2313},ISBN = {9781400881420},}[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(March1978), pp.283–309.MR0482765Zbl0383.57006article
Let\( \mathcal{H}(n) \) be the group of orientation-preserving self-homeomorphisms of a closed oriented surface\( \operatorname{Bd} U \) of genus\( n \), and let\( \mathcal{H}(n) \) be the subgroup of those elements which induce the identity on\( H_1(\operatorname{Bd} U;\mathbf{Z}) \). To each element\( h \in \mathcal{H}(n) \) we associate a 3-manifold\( M(h) \) which is defined by a Heegard splitting. It is shown that for each\( h\in\mathcal{H}(n) \) there is a representation\( \rho \) of\( \mathcal{H}(n) \) into\( \mathbf{Z}/2\mathbf{Z} \) such that if\( k\in\mathcal{H}(n) \), then the\( \mu \)-invariant\( \mu(M(h)) \) is equal to the\( \mu \)-invariant\( \mu(M(kh)) \) if and only if\( k\in\operatorname{kernel} \rho \). Thus, properties of the 4-maniolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from\( \mathcal{H}(n) \) onto\( \mathbf{Z}/2\mathbf{Z} \) are studied and are shown to constitute a complete conjugacy class of subgroups of\( \mathcal{H}(n) \). The class has nontrivial finite order.
| Robert Francis Craggs | Related |
@article {key0482765m,AUTHOR = {Birman, Joan S. and Craggs, R.},TITLE = {The \$\mu\$-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold},JOURNAL = {Trans. Am. Math. Soc.},FJOURNAL = {Transactions of the American Mathematical Society},VOLUME = {237},MONTH = {March},YEAR = {1978},PAGES = {283--309},DOI = {10.2307/1997623},NOTE = {MR:0482765. Zbl:0383.57006.},ISSN = {0002-9947},}[5]J. S. Birman and X.-S. Lin:“Knot polynomials and Vassiliev’s invariants,”Invent. Math.111 : 2(1993), pp.225–270.MR1198809Zbl0812.57011article
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.
| Xiao-Song Lin | Related |
@article {key1198809m,AUTHOR = {Birman, Joan S. and Lin, Xiao-Song},TITLE = {Knot polynomials and {V}assiliev's invariants},JOURNAL = {Invent. Math.},FJOURNAL = {Inventiones Mathematicae},VOLUME = {111},NUMBER = {2},YEAR = {1993},PAGES = {225--270},DOI = {10.1007/BF01231287},NOTE = {MR:1198809. Zbl:0812.57011.},ISSN = {0020-9910},CODEN = {INVMBH},}[6]J. S. Birman and M. D. Hirsch:“A new algorithm for recognizing the unknot,”Geom. Topol.2(1998), pp.175–220.MR1658024Zbl0955.57005article
The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example isnot the unknot.
| Michael David Hirsch | Related |
@article {key1658024m,AUTHOR = {Birman, Joan S. and Hirsch, Michael D.},TITLE = {A new algorithm for recognizing the unknot},JOURNAL = {Geom. Topol.},FJOURNAL = {Geometry and Topology},VOLUME = {2},YEAR = {1998},PAGES = {175--220},DOI = {10.2140/gt.1998.2.175},NOTE = {MR:1658024. Zbl:0955.57005.},ISSN = {1465-3060},}[7]J. S. Birman and W. W. Menasco:“Stabilization in the braid groups, I: MTWS,”Geom. Topol.10 : 1(2006), pp.413–540.MR2224463Zbl1128.57003article
Choose any oriented link type\( \mathscr{X} \) and closed braid representatives\( X_+ \),\( X_- \) of\( \mathscr{X} \), where\( X_- \) has minimal braid index among all closed braid representatives of\( \mathscr{X} \). The main result of this paper is a ‘Markov theorem without stabilization’. It asserts that there is a complexity function and a finite set of ‘templates’ such that (possibly after initial complexity-reducing modifications in the choice of\( X_+ \) and\( X_- \) which replace them with closed braids\( X^{\prime}_+ \),\( X^{\prime}_- \)) there is a sequence of closed braid representatives\[ X^{\prime}_+ = X^1 \to X^2 \to \cdots \to X^r = X^{\prime}_- \] such that each passage\( X^i\to X^{i+1} \) is strictly complexity reducing and non-increasing on braid index. The templates which define the passages\( X^i \to X^{i+1} \) include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index\( m\geq 4 \) a finite set\( \mathscr{T}(m) \) of new ones. The number of templates in\( \mathscr{T}(m) \) is a non-decreasing function of\( m \). We give examples of members of\( \mathscr{T}(m) \),\( m\geq 4 \), but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.
| William Wyatt Menasco | Related |
@article {key2224463m,AUTHOR = {Birman, Joan S. and Menasco, William W.},TITLE = {Stabilization in the braid groups, {I}: {MTWS}},JOURNAL = {Geom. Topol.},FJOURNAL = {Geometry and Topology},VOLUME = {10},NUMBER = {1},YEAR = {2006},PAGES = {413--540},DOI = {10.2140/gt.2006.10.413},NOTE = {MR:2224463. Zbl:1128.57003.},ISSN = {1465-3060},}[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.MR2255503Zbl1130.57005ArXivmath.GT/0310280article
The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.
| William Wyatt Menasco | Related |
@article {key2255503m,AUTHOR = {Birman, Joan S. and Menasco, William W.},TITLE = {Stabilization in the braid groups, {II}: {T}ransversal simplicity of knots},JOURNAL = {Geom. Topol.},FJOURNAL = {Geometry and Topology},VOLUME = {10},NUMBER = {3},YEAR = {2006},PAGES = {1425--1452},DOI = {10.2140/gt.2006.10.1425},NOTE = {ArXiv:math.GT/0310280. MR:2255503. Zbl:1130.57005.},ISSN = {1465-3060},}