At the end of the summer in 1975 I moved from southern California to the Bay Area to spend the next six years going through the doctoral program in mathematics at UC Berkeley. I now realize that that time interval was a unique slice of mathematical history: almost all areas of low-dimensional topology were accessible. As I peruse again the 1980s edition of Kirby’s problem list, I’m struck by the fact that almost all of the questions were understandable, at least in a rudimentary way. Additionally, the mathematical culture that I was immersed in was unique in time and place. I was amongst an extraordinary grouping of fellow students — Cameron Gordon, , , , and , among others. The existence of this mathematical culture was due to the efforts and energy of Rob Kirby., , , , , , to name a few — and we had contact with such visting scholars as
I had graduated from UCLA the previous spring, and had turned down an acceptance-with-funding offer into their PhD program. Without a similar funding offer from Berkeley, I had to sell a graduation present — a used 1962 Mercedes Benz — to finance my first year of graduate school. It was a gamble, and I wondered at the time whether it would “pay off”. Because of two events that occurred during the first week of classes, I concluded that I’d made the right choice.
In fact, the first event occurred the first day of classes. Walking down the hallway between a class break on the first floor of Evans Hall, I heard someone call out my name, “Menasco!”. Turning around I saw Chuck Livingston who I had not seen for two years. At UCLA we had gone through honors calculus and linear algebra together during our sophomore year, and then he had left California, transferring to M.I.T. He looked exactly the same except for the fact that he had traded in a crew-cut for a ponytail. After some catching up, we found out that we were both interested in geometry/topology.
The second event occurred upon my first visit to the Evans Hall 10th floor commons room around tea time. There I saw a sequence of large drawings of links diagrams, working out what I would later recognize as Kirby calculus calculations. My initial impression was, “Wow, someone gets paid to draw these things!” I was hooked. Topology was the direction to go.
I did not fall solidly within Rob’s cultural orbit until the beginning of my third year at Berkeley when I walked into Rob’s office and received the standard Kirby reply when I asked if he would be my thesis advisor. He pointed me to a couple of stacks of manuscripts and told me to pick some and start reading them. Close to the top of the stacks were papers about Thurston’s work on surface foliations and the Thurston norm. Later that fall I gave a couple of talks in Rob’s regular Wednesday afternoon graduate seminar, trying to communicate an understanding of Thurston’s completion of Teichmüller space and the unit ball for the Thurston norm for the Whitehead link [e4]. Thanks to some coaching by Bill Goldman and Thurston himself during one of his west coast visits, these initial showings were not a total disaster. Moreover, I was sold on the interplay between topology and geometry in low-dimensional hyperbolic manifolds. It was time to move on to Thurston’s massive work, The Geometry and Topology of Three-Manifolds [e6].
What I remember most about working through Thurston’s “magnum opus” is Joel Hass, Rob and me sitting in Rob’s office, all silently reading — a struggle where progress was measured through line-by-line understanding. The initial head-scratcher was Thurston’s description of the figure-eight knot complement as being the union of two ideal tetrahedra. The reader’s main clue was a large illustration of the figure-eight draped over the exterior of a tetrahedron. This was the first significant problem that I decided to tackle on my own: how to describe this construction for the figure-eight complement in a “transparent” way, and how to generalize it to other knot complements.
Before the age of personal computers equipped with flexible graphics programs, drawing detailed examples could be tedious. I sped up the process of creating a library of examples by means of the department’s Xerox machine. Rob’s contribution to this effort was to give me his copying-machine PIN code — and, hence, an unlimited number of Xerox copies. After about a month’s worth of examples, I had a construction which generalized Thurston’s initial examples: a decomposition of alternating knot complements in the ideal polyhedrons. So back to Rob’s Wednesday seminar where I described the construction. At the time Rob was teaching a course on 3-manifolds; there, he gave a more transparent description of my “transparent” construction (see [e1]).
Some time after this, Cameron Gordon visited Berkeley. I managed to buttonhole him in the tea room to show him my construction on one of the blackboards. I’m not sure how much of my blackboard graphics he understood, but he did ask a question that sent me on a different course: “Can you show that every torus in the complement of an alternating link has a meridian curve?” Well, I didn’t even know that was an interesting question. One of the big benefits of such a rich culture was having dignitary-topologists popping in and out, raising all the questions you didn’t know were there. I think it took me less than two days to see how to argue that a surface in normal position with respect to the polyhedron decomposition of an alternating knot complement always had a meridian curve. Back to Rob’s Wednesday’s seminar with this one.
Here my recollection gets a little bit fuzzy. I believe that by the time I got back to the Wednesday seminar to present the meridian curve result Larry Seibenmann had showed up in the department. He sat in on my talk with all the arguments about having to put the surface in normal position with respect to the polyhedron decomposition, and in the end said: “This will not do. You have to make it simpler.” My impression of Rob’s reaction was “Larry’s right.”
So back to the drawing board. I got rid of the polyhedron structure, threw in some crossing balls in the projection of an alternating knot, and figured out how to define normal position for a surface with respect to just the knot projection. By the time I was ready to go back to Rob’s seminar it was summer and Ray Lickorish showed up for a visit. I gave a talk where I proved that an alternating knot is prime if and only if it is obviously prime. After my talk Ray told Rob that “this was really good stuff”. Rob’s reply was, “Well, it’s getting there.”
It was thanks to Larry and Ray that I found out about the Tait conjectures, and, more specifically, the Tait flyping conjecture. Again, questions and problems I had never heard of. I would end up spending the next 10 years chasing after the flyping conjecture (see [e7]).
Towards the end of my graduate years at Berkeley, Allen Hatcher was in town on an extended visit and taught a course in low-dimensional topology. At some point I started pestering him in his office, showing him the ideal polyhedron decomposition and the normal form for surfaces in alternating knot complements with respect to their reduced alternating projections. During these sessions, another question came up that I had never even thought about: “Does an incompressible surface in the complement of an alternating knot survive after nontrivial Dehn surgery?” I don’t know how many false proofs Allen shot down in his office. At some point I did feel like I had over stayed my welcome. But, one day we had success. A simple diagram on the board in his office illustrated that an incompressible surface in the complement of a knot with two nonisotopic meridian curves survived all nontrivial Dehn surgeries. Later this argument would be generalized by [e3] and would show up as part of the argument in the cyclic surgery theorem (see [e5]).
My indebtedness to Allen is hard to overstate. Without his intervention my first paper in Topology [e2] would never have seen print.
“I am a Kirby student.” This is a standard reply I have given over the years to the question of who was my thesis advisor. But, I now view this reply as a culture-of-origin statement, not a student/teacher declaration. Although it may seem that I am the prominent character in the above narrative, the major protagonist is in fact the rich culture created by Rob. Having been given the privilege and honor of being a coeditor on Rob’s Celebratio volume, I have discovered my experience was a common one in my peer cohort — Joel Hass, Danny Ruberman, Chuck Livingston, and others. I don’t know exactly how he did it since I was never privy to his behind-the-scenes workings, but I am very thankful for it. I do not think Rob ever read my doctoral thesis. He didn’t need to. Doing so would have been superfluous. The pressures, corrections, and value-added functions of the “Kirby culture” were sufficient.
William W. Menasco received his Ph.D. from UC Berkeley in 1981 under the direction of Rob Kirby. After a postdoc at Rugters University, he joined the faculty at the University at Buffalo–SUNY, where he is Professor of Mathematics.