Celebratio Mathematica

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 — Joel Hass, Danny Ruber­man, John Harer, Tim Co­chran, Chuck Liv­ing­ston, Bill Gold­man, to name a few — and we had con­tact with such vist­ing schol­ars as Camer­on Gor­don, Larry Sieben­mann, Ray Lick­or­ish, Al­len Hatch­er, and Bill Thur­ston, among oth­ers. The ex­ist­ence of this math­em­at­ic­al cul­ture was due to the ef­forts and en­ergy of Rob Kirby.

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 Ham­ish Short [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.