Celebratio Mathematica

I met Wal­ter when I ar­rived to take up a Lec­turer po­s­i­tion at Ohio State Uni­versity in Septem­ber 1987. I was ap­poin­ted as a Lec­turer rather than an In­struct­or be­cause I would not de­fend my Ph.D back in Ab­er­deen un­til March 1988. I was ap­poin­ted as an In­struct­or in 1988 and re­mained at Ohio State un­til June 1990. It is an un­der­state­ment to say that meet­ing Wal­ter and work­ing closely with him over those three years had a massive im­pact on my math­em­at­ics and ca­reer. I will com­ment on some of our joint work be­low. In­deed, look­ing back at my time at Ohio State, it was an in­cred­ibly ex­cit­ing peri­od: along with Wal­ter, Dan Burghelea, Ruth Char­ney, Mike Dav­is (who is still there), Henry Glover, and Guido Mis­lin, were all on the fac­ulty then; Tadeusz Januskiewicz was also vis­it­ing; and Mike’s former stu­dent Gabor Mous­sang was work­ing on his Ph.D thes­is. In ad­di­tion to the people in geo­metry and to­po­logy, Avn­er Ash was there as part of a strong num­ber the­ory group, and so Ohio State was alive in math­em­at­ic­al areas that I was very much in­ter­ested in.

A par­tic­u­lar per­son­al high­light of my time at Ohio State is that I proved that the fig­ure-eight knot is the only arith­met­ic knot [e2]. I had star­ted think­ing about this when I was a gradu­ate stu­dent, had sev­er­al false proofs, and routinely would show up in Wal­ter’s of­fice in Cockins Hall (ad­ja­cent to the old math build­ing at Ohio State) with more false proofs. Then, one day I went to Wal­ter’s of­fice, and we star­ted to chat about a pa­per by Gonza­lez-Acuña and Whit­ten [e1] that I was read­ing, and as I re­call, Wal­ter was ex­plain­ing some part of the proof of a lemma in that pa­per, when it sud­denly star­ted to be­come clear that this lemma po­ten­tially could be tweaked to help in the proof. In­deed this turned out to be the case, and it be­came one of the key points in the proof that the fig­ure-eight knot was the only arith­met­ic knot. In hind­sight I can’t ima­gine this hap­pen­ing without Wal­ter’s un­end­ing pa­tience with me show­ing up in his of­fice to talk about the fig­ure-eight and the math­em­at­ic­al dis­cus­sions that we had around this prob­lem over the course of many months.

Wal­ter was in­cred­ibly gen­er­ous with his time and ideas, both in his of­fice and at home, where Anne and Wal­ter of­ten had me over for din­ner. I must also give cred­it to Wal­ter for in­tro­du­cing me to Sam Adams, a beer largely un­known back in the late 1980s out­side of Bo­ston, but it was a “life saver” over the usu­al se­lec­tion of beers avail­able back then!

As is well known Wal­ter has broad tastes in math­em­at­ics, and this can be said even amongst his work on 3-man­i­folds and geo­met­ric group the­ory. I will com­ment on my work with Wal­ter be­low (and some oth­er of Wal­ter’s re­lated work) but I will briefly men­tion some per­son­al high­lights of his oth­er work. This will not do justice to the breadth and depth of Wal­ter’s work, and there are many more art­icles that I could com­ment on; I hope these oth­ers will be ad­dressed by col­leagues.

Fore­most amongst these is his pa­per with Za­gi­er [3] which provides a beau­ti­ful ac­count of Thur­ston’s Dehn Sur­gery the­or­em, in­clud­ing quant­it­at­ive state­ments about change in volume and how the length of a core geodes­ic var­ies. An­oth­er pa­per of Wal­ter’s in 3-man­i­folds that I ad­mire and view as im­port­ant is the pa­per [1] (with D. Eis­en­bud and U. Hirsch). This pa­per gives cri­ter­ia for a Seifert 3-man­i­fold to ad­mit fo­li­ations whose leaves are all trans­verse to the fibers. The cri­ter­ia are nu­mer­ic­al in­equal­it­ies that gen­er­al­ize the cri­ter­ia of Mil­nor and Wood. There has been re­newed in­terest in this pa­per lately due to its rel­ev­ance to the L-space con­jec­ture. Wal­ter’s Bran­de­is lec­ture notes [2] are beau­ti­fully writ­ten and one of the go-to texts to learn about Seifert man­i­folds (and which I con­tin­ue to use even now). Fi­nally, Wal­ter’s pa­per [4] (with R. Bieri and R. Strebel) can be viewed as a far reach­ing gen­er­al­iz­a­tion of Thur­ston’s norm ball for 3-man­i­fold groups, and is an­oth­er of Wal­ter’s pa­pers which con­tin­ues to have long term im­pact.

My Ph.D thes­is, “Arith­met­ic Klein­i­an groups and their Fuch­sian sub­groups”, was largely fo­cused on arith­met­ic sub­groups of \( \operatorname{PSL}(2,\mathbb{C}) \). To­wards the end of my Ph.D, I had gradu­ally star­ted to broaden my view­point and think about broad­er num­ber the­or­et­ic ap­plic­a­tions in the study of the geo­metry and to­po­logy of hy­per­bol­ic 3-man­i­folds. However, I was really un­sure as to wheth­er really: (a) it was worth think­ing about, and (b) would any­body else care. Wal­ter quickly re­af­firmed both of these points to me, and this formed the plat­form for our work and some of my oth­er work com­pleted or star­ted at Ohio State. Re­gard­ing (b) above, it was clear from when I ar­rived at Ohio State that Wal­ter had already thought about these num­ber the­or­et­ic con­nec­tions more broadly; per­haps this was un­der­stand­able giv­en that his Ph.D was co-ad­vised by Hirzebruch (who had a keen in­terest in con­nec­tions of num­ber the­ory and to­po­logy) and he had Za­gi­er as a co-au­thor. In­deed, at the time of my ar­rival, Wal­ter was very much in­ter­ested in the Bloch group of num­ber fields and its re­la­tion to tri­an­gu­la­tions of 3-man­i­folds.

I have writ­ten five pa­pers with Wal­ter, the last one in 2007 also joint with C. Lein­inger and D. B. McReyn­olds (these are lis­ted be­low), and com­pleted sev­en more whilst at Ohio State that were all greatly in­flu­enced by Wal­ter. I will com­ment in a little more de­tail on one of my pa­pers [7] with Wal­ter, and the book To­po­logy ’90 [6] (ed­ited with B. Apanasov and L. Sieben­mann).

This pa­per es­sen­tially crys­tal­lized many of the ideas that Wal­ter and I talked about re­gard­ing con­nec­tions between num­ber the­ory and hy­per­bol­ic 3-man­i­folds. In par­tic­u­lar, we in­tro­duce the nota­tion in­vari­ant trace-field and in­vari­ant qua­ternion al­gebra that have since be­come stand­ard. I will com­ment on two the­or­ems in this pa­per. First, we give a geo­met­ric in­ter­pret­a­tion of the in­vari­ant trace-field, namely:

Apart from be­ing very pleas­ing, Wal­ter was in­ter­ested in this res­ult from his work on the Bloch group and com­put­ing the Chern–Si­mons in­vari­ant from (ideal) tri­an­gu­la­tions of 3-man­i­folds. This is evid­ent in both his in­di­vidu­al work on these top­ics (see [9], [13] and [16]) and in his joint work with J. Yang [11], [14] (I will de­fer to oth­er com­ment­ar­ies for fur­ther de­tails on this top­ic).

Wal­ter en­thu­si­ast­ic­ally em­braced com­puter ex­per­i­ment­a­tion in his work, and when I ar­rived at Ohio State, he already had his own pro­gram (writ­ten in UBasic as I re­call) that al­lowed ex­per­i­ment­a­tion with Dehn sur­ger­ies on knots. In par­tic­u­lar, be­ing a schol­ar Wal­ter was fa­mil­i­ar with the LLL al­gorithm, and this al­lowed him to make ex­act com­pu­ta­tions of trace-fields (and in­vari­ant trace-fields) of Dehn sur­ger­ies on knots. Back then this seemed to me (a novice then and now in im­ple­ment­ing com­pu­ta­tion­al meth­ods) quite mi­ra­cu­lous, but this gave us large (back in the day) num­bers of ex­amples with which to try to un­der­stand trace-fields of knots, and how they changed un­der Dehn sur­gery. This was the pre­curs­or to the pro­gram Snap, writ­ten when Wal­ter moved to Mel­bourne along with D. Coulson, O. Good­man and C. Hodg­son (see [15] for a dis­cus­sion of this pro­gram). Roughly speak­ing, this won­der­ful pro­gram blends the pro­gram SnapPy (for ex­plor­ing hy­per­bol­ic struc­tures on 3-man­i­folds) with Pari (for ex­plor­ing num­ber fields amongst oth­er things), to al­low for ex­act com­pu­ta­tion of arith­met­ic in­vari­ants of hy­per­bol­ic 3-man­i­folds.

An­oth­er res­ult we proved in [7], and a con­jec­ture we put for­ward there (ac­tu­ally this came a little later, but it can be traced to this work), con­cerns prop­er­ties of com­men­sur­ab­il­ity classes of knot com­ple­ments. Be­fore com­ment­ing on this work, I would like to note that Wal­ter re­tained a keen in­terest in the prop­erty of com­men­sur­ab­il­ity in many dif­fer­ent as­pects of his math­em­at­ic­al in­terests; see for ex­ample the pa­pers [19], [12], and, in a slightly dif­fer­ent con­text, [20]

Re­turn­ing to our work on com­men­sur­ab­il­ity, we proved the fol­low­ing. For con­text, hav­ing re­cently proven that the fig­ure-eight knot is the unique arith­met­ic knot, Wal­ter and I were in­ter­ested in un­der­stand­ing com­men­sur­ab­il­ity classes of knot com­ple­ments. By the work of Mar­gulis, if the knot \( K \) is non­arith­met­ic, then \( S^3\setminus K=\mathbb{H}^3/\Gamma \) cov­ers a unique min­im­al or­bi­fold \( Q \), which arises as the quo­tient of \( \mathbb{H}^3 \) by the com­men­sur­at­or of \( \Gamma \). The com­men­sur­at­or is eas­ily seen to con­tain the nor­mal­izer, and we defined a hy­per­bol­ic knot oth­er than the fig­ure-eight knot to have hid­den sym­met­ries when the com­men­sur­at­or of its fun­da­ment­al group is lar­ger than its nor­mal­izer. We proved the fol­low­ing res­ult.

To say that \( Q \) has a ri­gid cusp means that the cusp end of \( Q \) has the form \( B\times [0,\infty) \) where \( B=\mathbb{E}^2/D \) and where \( D \) is the ori­ent­a­tion-pre­serving sub­group of the group gen­er­ated by re­flec­tions in one of the Eu­c­lidean tri­angles \( \Delta(2,4,4) \), \( \Delta(3,3,3) \) and \( \Delta(2,3,6) \). In a sep­ar­ate art­icle in To­po­logy ’90, I. Aitchis­on and H. Ru­bin­stein [e3] con­struc­ted the so-called do­deca­hed­ral knots that are ex­amples of knots with hid­den sym­met­ries. Based on this and some ex­per­i­ment­a­tion we raised the fol­low­ing ques­tion.

Hav­ing hid­den sym­met­ries seems ex­tremely rare, and with the pas­sage of time, it seemed reas­on­able to up­grade the ques­tion Wal­ter and I posed to the status of a con­jec­ture: the only non­arith­met­ic hy­per­bol­ic knots which ad­mit hid­den sym­met­ries are the two do­deca­hed­ral knots. Hid­den sym­met­ries of knots have ap­peared in vari­ous places since our pa­per; per­haps most in­ter­est­ingly and most closely re­lated to our work is in the pa­per [e4], which es­tab­lished that for knots without hid­den sym­met­ries there are at most 3 hy­per­bol­ic knot com­ple­ments in a com­men­sur­ab­il­ity class.

An­oth­er as­pect of my time spent at Ohio State with Wal­ter — and one which I feel very for­tu­nate to have been in­volved with — was our or­gan­iz­a­tion of a Re­search Semester in Low Di­men­sion­al To­po­logy, Feb­ru­ary — June 1990. This took place un­der the aus­pices of the re­cently cre­ated In­ter­na­tion­al Math­em­at­ic­al Re­search In­sti­tute at Ohio State, and was the first ma­jor pro­gram of the In­sti­tute. The main top­ics of the Re­search Semester in­cluded many (un­sur­pris­ingly) that were close to Wal­ter’s in­terests: the geo­metry and to­po­logy of 3-man­i­folds, with par­tic­u­lar em­phas­is on hy­per­bol­ic 3-man­i­folds and their in­ter­ac­tions with num­ber the­ory, in­vari­ants of 3-man­i­folds re­lated to quantum field the­ory and plane al­geb­ra­ic curves. There were also a few talks in geo­met­ric group the­ory, rep­res­ent­ing Wal­ter’s grow­ing in­terest in the field. As a young in­struct­or, hav­ing all these vis­it­ors around and a vi­brant (usu­ally weekly) sched­ule of talks was in­cred­ibly ex­cit­ing. The book To­po­logy ’90 [6] con­tained con­tri­bu­tions from par­ti­cipants in the Re­search Semester.

As I stated at the out­set, I feel enorm­ously grate­ful to have spent three years with Wal­ter at the same in­sti­tu­tion, to have vis­ited him over the years on his travels, to have been his col­league, to have be­nefited im­mensely from his ment­or­ship and his math­em­at­ic­al taste and vis­ion, and to have been able to call him a friend for these last 35 (!) years.

Alan Re­id re­ceived his PhD in 1988 from the Uni­versity of Ab­er­deen. After a num­ber of postdoc­tor­al po­s­i­tions in the UK and USA, he was ap­poin­ted as an As­sist­ant Pro­fess­or at UT Aus­tin in 1996 and re­mained on the fac­ulty at UT un­til 2017 when he joined the De­part­ment of Math­em­at­ics at Rice Uni­versity as the Edgar Odell Lovett Chair and Pro­fess­or of Math­em­at­ics. He is at present Chair of the de­part­ment.