A PERSONAL ACCOUNT OF THE DISCOVERY OF HYPERBOLIC STRUCTURES ON SOME KNOT COMPLEMENTS 3 ROBERT RILEY 1 Mathematical Sciences, Binghamton University, Binghamton, NY, USA 0 2 n MSC classification (2010): a J Primary: 57M50, 57M25, 9 Secondary 30F40, 01A60 1 ] Keywords: Hyperbolic structures, Knot complements O H Abstract: I give my view of the early history of the discovery of hyper- . bolic structures on knot complements from my early work on represen- h tations of knot groups into matrix groups to my meeting with William t a Thurston in 19761 m [ 1. Introduction 1 v 1 I discovered, quite unexpectedly, the phenomenon of hyperbolic struc- 0 ture on three knot complements early in 1974, and managed to get two 6 papers on the topic published in 1975. At some moment between the 4 . dates of publication of these papers, William Thurston independently 1 0 discovered the phenomenon and ran away with the idea. In late June 3 or early July 1976 he learned of my work, and so when we met later in 1 July he immediately told me that he had been trying for about a year : v to prove the hyperbolization conjecture for Haken 3–manifolds. i X Colin Adams published a semipopular account of knot theory in “The r a Knot Book” [1], and a copy of this came into my hands recently. On page 119 he gives an account of the hyperbolic structure discovery which is just plain wrong.2 He does get the names of the two people 1ThisarticlewaswrittenbyRobertRileyabouttenyearsbeforehisdeathin2000 and never submitted for publication. An explantion of why it is being published now and some information about Riley and this article is given in [BJS] which accompanies this article in this issue of the journal. 2Riley refers here to the first edition of The Knot Book, published in 1994. In §5.3 of the 2004 edition, published by the AMS, there is a concise, corrected account of this discovery, together with an excellent elementary introduction to hyperbolic knots. 1 2 ROBERT RILEY concerned and the priority right, but nothing else. The present paper is an attempt to set the record straight. I shall relate what I did, why, and when. There will be too much detail about small matters, but this will convey the spirit of my projects. Indeed, I think my old papers were very open about my project, and a close look at them and their dates of submission should have made the present history unnecessary. Furthermore, Bill Thurston’s account of my work in [13] is entirely fair, except for being too generous about my influence on his thinking. So below I give the history of my project from its beginning to the moment I met Professor Thurston. The story is told as I saw it, and the emphasis is on motivation and dates. Many dates are only approximate because most entries in my notebooks are undated, but the uncertainties are never more than about a month. I include an intermediate example, worked out between the discoveries of the hyperbolic structures for the figure–eight knot (4 ) and for 5 . This example ought not be on the 1 2 main line of development, but in fact it was, and it served to undermine my initial expectation that the figure–eight is the only knot which could possibly be hyperbolic. I close with some comments on the early work of H. Gieseking and Max Dehn, and on the article [15] of W. Thurston. 2. The early years On settling in Amsterdam in October 1966 I wrote off to virtually everyone publishing in knot theory for their reprints and preprints. I recall with gratitude that R.H. Fox and H. Seifert were especially generous. An unassuming little paper by Fox [2] written in Utrecht some 20 miles away, took my fancy. Here Fox advertised the notion of longitude in a knot group by using it, together with representations on the alternating group A , to distinguish the square and granny knots. 5 I was intrigued by the success of A , and took the first steps toward 5 writing out explicit procedures to find all A –representations of a knot 5 group in 1967–68. When I got my first temporary appointment at Southampton (England) in 1968 this became my main project, with results summarized in [6, 7]. So by 1970 I was after the parabolic repre- sentations (p–reps) of a knot group, initially because they were easier to manage than the general non–abelian representations (nab–reps). The 2–bridge case is especially tractable, because the representations are governed by a simple polynomial whose rule of formation is easily programmed in Fortran. This tractability extends to all r–bridge knots which are symmetric about an r–fold axis of rotation that cyclically permutes the bridges, but most knots of bridge number > 2 are not THE DISCOVERY OF HYPERBOLIC STRUCTURES 3 so symmetric. The explicit algebraic description of the equivalence classes of p–reps of an unsymmetric knot is so difficult that only a few examples have been worked explicitly, and I have found the full curve of all nab–representations of only one unsymmetric 3–bridge knot, 8 . Around 1971 I wrote some primitive Fortran programs to find the 20 p–reps of a few 3–bridge knots and used the output to discover the commuting trick of [7], [11, II], but at the time this topic was mainly pure frustration. In 1971 a plea for help from me was passed on to Professor G.E. Collins, the instigator of the SAC–1 file of Fortran routines for doing the kind of algebraic calculations I needed. He sent me a pile of very poorly printed manuals containing the program listings, lots of errata slips, and the advice that the 24 bit word size of the Southampton University computerwouldrequiresomedoublyrecursiveprogramminginassembly language. (The reference count field in a SAC atom would be too small without this recursion, and hence impose a strict limit on the allowed complexity of calculations). He also mentioned that I would need to get someone to punch up the 6000 or so cards of the 1971 SAC. Well, that someone had to be me, but fortunately only some 4000 cards, plus the assembly language parts, were needed for my application. It took about eight months to do all this, and I never did get the double recursion for the reference count right. So my more ambitious calculations were killed as soon as the reference count tried to reach 128, but I still managed to do most of what I wanted. By 1 October 1972, the day my fourth temporary appointment at Southampton ceased, I had done the elimination–of–variables part of the solving for an algebraic description of the set of p–reps for several 3–bridge knots, including 9 . Each 35 SAC run required several hours of CPU time, and could not have been attempted during term time. Perhaps some distorted memory of this story is the source of the “immense computer program that was designed to attempt to show that some knots are hyperbolic” bit in Adams’ account. In fact, the PNCRE package which does just this was developed from 1976, and it was always fast enough for term time, even during the day on a grossly overloaded 1960’s computer. 3. The preparation In October 1972 I had a large pile of SAC output which needed more computer analysis to become meaningful, and no prospect of further employment. So I spent the next three months walking the Pennine Way and walking in Wales until the prospect of a six month appointment 4 ROBERT RILEY in Strasbourg opened up. While I was walking in the Vosges this materialized, and I was able to complete the algebraic description of the equivalence classes of p–reps for several knots, including 9 , cf. [11]. 35 (I recall a puzzling difficulty with 9 that was explained a decade later 32 as the consequence of dropping the deck of data cards, perhaps in 1971, and reassembling it almost exactly right). The knot 9 has a large symmetry group (dihedral of order 12, [11]), 35 and also an unusually large number of algebraic equivalence classes of p–reps, facts which I believe are related. The SAC calculations had given me a polynomial p(x) ∈ Z[x] of degree 25 which I had to factor as the first step. When one has no symbolic manipulation package available this is done by finding the roots of p(x) = 0 and examining them for clues. The polynomial p(x) (and its relative for 9 which 48 was even worse) defeated several commercially produced root–finding routines, but a final resort routine succeeded, sort of, and I was able to infer factors p = 1+x, p = 1+2x+7x2 +5x3 +x4, p = ··· , 1 2 3 and soon p(x) = (1+x)10p (x)2··· . 2 Only the cubic factor remained unguessed, and of course it turned out to be the one giving the hyperbolic structure four years later. Each factor p (x) of p(x) had to be tested to see if it gave an equivalence k class of p–reps or was spurious, and I expected 1+x to be spurious. To my surprise it gave p–reps on (cid:28)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:29) (cid:20) (cid:21) 1 1 1 0 1 0 1 i G = , , A A−1 ⊂ SL (Z[i]), A = , i 0 1 −1 1 i −1 1 i 2 i 0 1 √ where i = −1. This was in June 1973, and I probably did not understand what a Kleinian group is at the time, but I could see G is i discrete and wondered what its presentation was. Also, as I watched the printout emerge from the line printer I guessed that these p–reps must be an instance of an undiscovered theorem, and the same evening stated and proved the theorem. (Writing it up for publication is taking longer. In December 1991 I used Maple to extend the theorem to algebraic varieties of nab–reps and add some new material. In 1993 I told Tomotada Ohtsuki about this, giving no detail, and he promptly found a better proof and more new material. I hope to proceed to a joint paper soon.) After the summer vacation of 1973 when I returned to Southampton, the professors of the mathematics department granted me the use of THE DISCOVERY OF HYPERBOLIC STRUCTURES 5 an office and all university facilities, except the computer which was heavily overloaded. By then I had learned by some osmosis what a Kleinian group is and read Maskit’s paper [4] on Poincar´e’s Theorem on Fundamental Polyhedra. This made progress on G above possible, i and I soon had its presentation. (I also found that Fricke and Klein had considered G , or something very like it, cf. Fig. 151 on page 452 i of [3].) Success with G led to success with the image πKθ of a p–rep i of the figure–eight knot group in November 1973. Recall that √ (cid:28)(cid:20) (cid:21) (cid:20) (cid:21)(cid:29) 1 1 1 0 −1+ −3 πKθ = , , ω = , 0 1 −ω 1 2 so the group is obviously discrete and only its presentation was in doubt. I remember my surprise at finding this p–rep is faithful. The first version of my account [8] of this was received by the Editors on 30 November 1973, and it didn’t mention the orbit space H3/πKθ because I had not even thought of it. Why not?! Well, the result was perhaps a fortnight old, and I didn’t have a premonition of hyperbolic structure on knot complements. Years later I learned that it had not only been thought of, but attempted and discussed privately by the Kleinian groupies since 1968. Nothing had been written and none of this had reached me. The key to seeing that the orbit space of πKθ had to be the figure–eight complement was seeing the peripheral torus in the orbit space. This torus occurs as the image of Euclidian plane Π(t) = {(z,t) : z ∈ C} ⊂ H3 for any t > 1. In my diagram Π(t) meets the fundamental domain not in a parallelogram but in a zigzag shape (four hexagonal discs), and perhaps the zigzag temporarily prevented me from seeing the torus. This is silly, because the stabilizer of the torus is the free abelian group (πKθ) generated √ ∞ by z (cid:55)→ z + 1, z (cid:55)→ z + 2 −3, and (πKθ) has to be considered ∞ explicitly during the verification that Poincar´e’s theorem applies to my supposed Ford fundamental domain. But silly or not, it took perhaps seven weeks, till January 1974, for me to see the torus. Verification that H3/πKθ = S3 −fig–eight took perhaps a day, and consisted of looking at my reprint of Waldhausen’s paper [16]. It seems unfortunate that this was too easy, and that I should have been forced to develop a direct geometrical argument, but once the pressure was off I didn’t want to do it. I expect that a direct geometrical construction works for all non–torus two bridge knots, and that it would prove the conjectures of [12, §4], so the matter will not be a waste of effort. 6 ROBERT RILEY The figure–eight discovery was not decisive for me as it was for Thurston. I expected that Shimizu’s lemma, viz. (cid:28)(cid:20) (cid:21) (cid:20) (cid:21)(cid:29) 1 1 a b , is not discrete when ad−bc = 1, 0 < |c| < 1, 0 1 c d would preclude the discreteness of the images πKθ of the potentially faithful p–reps θ for all other knots. (In particular, I predicted Alan Reid’s theorem [5] that the figure–eight is the only arithmetic hyperbolic knot). However, by the time I mailed off the revised version of [8] that was actually accepted I had recognized the true situation, but, I suppose out of laziness, I didn’t revise [8] again to make an announcement. R.H. Fox died within a few days of the figure–eight discovery. 4. The intermediate example I now had a beautiful discovery, and a certain fear of testing whether something similar was true for the obvious next case, the knot 5 of two– 2 bridge types (7,3), (7,5). Instead of going for 5 directly I temporized 2 by taking up a different kind of example, the groups πKθ associated to a cubic factor f(u) of the p–rep polynomial for the knot 8 of 11 two–bridge types (27,17), (27,19). To give an account of this we need to recall the basics of two–bridge knot groups and their p–reps. A two–bridge knot normal form corresponds to a pair (α,β) of integers, where α > 1 is odd, β is odd, gcd(α,β) = 1, and −α < β < α. The knot group πK for (α,β) depends not on β itself but on |β|, so we may as well assume 0 < β < α. Then πK = |x ,x : wx = x w|, w = x(cid:15)1x(cid:15)2···x(cid:15)α−1, (3.1) 1 2 1 2 1 2 2 (cid:42) where (cid:15) = (cid:15) = ±1, and the exponent sequence (cid:15) = ((cid:15) ,··· ,(cid:15) ) j α−j 1 α−1 is determined by a simple rule, cf. [7, 12]. A longitude γ in the 1 peripheral subgroup (cid:104)x ,γ (cid:105) of x is a certain word w˜−1wx2σ on x ,x . 1 1 1 1 1 2 A normalized p–rep θ = θ(ω) of πK is a homomorphism such that (cid:20) (cid:21) (cid:20) (cid:21) 1 1 1 0 x θ = A = , x θ = B = B = , (3.2) 1 0 1 2 ω −ω 1 where ω ∈ C. Indeed, ω is a root of the p–rep polynomial Λ[u] ∈ Z[u] which may be reducible but which has no repeated roots. Then the longitude entry g(θ) or g(ω) for θ(ω) is found by (cid:20) (cid:21) −1 g(θ) γ θ = , 1 0 −1 THE DISCOVERY OF HYPERBOLIC STRUCTURES 7 and is readily computable once ω is known. To factor Λ(u) without a system like SAC, Macsyma, or Maple but when a polynomial root finding package is available, find the roots and list the pairs (ω,g(ω)). Factors stand out as having pairs where g(ω) evidently belongs to a proper subfield of Q(ω). In the case of 8 we found the factor 11 f(u) = −1+u(1+u)2 by g(θ) = −6 for its roots. The roots of f(u) are ω = −1.23278+0.79255i, ω = ω¯ , ω = 0.46557, (3.3) 1 2 1 3 (rounded to 5 decimal accuracy). Today this factor is explained as an instance of Theorem B of [12] and it clearly had something to do with the discovery of the theorem. I had f(u) by 1971. By February 1974 my worries about the figure–eight knot brought me to consider the group Γ = (cid:104)A,B(cid:105), B = B , where ω is the ω of (3.3). I ω 1 simply went for a Ford domain D of Γ using graph paper, compass and ruler, and the first programmable calculator available at Southampton. (That would have cost about two months gross salary if I had still been employed). It didn’t take long to get the diagram of Fig. 1, and when the time came to think about proof the closing trick and angle sum trick of [10] came to mind automatically. As far as I know this group Γ is the first group proved discrete by Poincar´e’s theorem where these tricks are necessary. Perhaps the first people to wonder about using Poincar´e’s theorem for computation with potentially discrete groups didn’t see these simple tricks in advance, didn’t have a specific example they really needed, and shied away from getting too involved. We give a little more detail on Γ and its Ford domain D illustrated in Fig. 1. This is taken from an unpublished paper CPG, written in late 1974 and early 1975, doing all the discrete non–Fuchsian cases where the group πKθ corresponds to a root of a cubic polynomial, viz. 5 , 7 , 2 4 and 8 . The case 5 is worked in [11], and 7 is similar to but easier 11 2 4 than 7 , also worked in [11] but much easier. 7 Let πK be the group of (27,17) presented as in (3.1), so Γ = πKθ as in (3.2). We have words u := x−1x x , v := ux−1x x−1, w := v x−1v−2x−1v x−1. 1 2 1 1 2 1 2 1 1 1 1 2 1 1 The word w of (3.1) is w x , so w x = x w holds in πK. These 1 1 1 1 2 1 words u, v were found by straightforward search of subsegments of w 1 to correspond to spheres carrying sides of tentative Ford domains. The search for the sides of a fundamental domain has to be guided by some 8 ROBERT RILEY THE DISCOVERY OF HYPERBOLIC STRUCTURES 9 principle, since a Cantorian exhaustion is too slow, and segments of w worked well, both here and later for all two bridge knots. We found easily that the elements A = x θ, U = uθ, V = v θ, W = w θ, V = U−1W 1 1 1 1 1 2 1 seem to be the side pairing transformations of the tentative Ford domain D of Fig. 1. Thus we read off from Fig. 1 a proposed presentation for Γ: generators A, U, V , V , W . relations 1 2 1 W2 = V3 = V3 = (A−1V )2 = (A−1V )2 = E, 1 1 2 1 2 V = W U−1, V = U−1W , U = A−1W AW A−1. 1 1 2 1 1 1 To use the closing trick and angle sum tricks of [10] it is necessary to verify directly that these relations hold in Γ. For this it helps to see copies of the modular group SL (Z) in Γ. Let 2 (cid:20) (cid:21) 1 u+u2 A := , ∗ 0 1 then (cid:20) (cid:21) (cid:20) (cid:21) 0 −1 1 −1 V ≡ A−1 A , V ≡ A A−1 (mod f(u)). 1 ∗ 1 1 ∗ 2 ∗ 1 0 ∗ So (cid:104)A,V (cid:105) and (cid:104)A,V (cid:105) are conjugate to SL (Z) in SL (Z[ω]). All the 1 2 2 2 proposed relations now can be verified by straightforward computation in SL (Z[u]) modulo f(u). Then the arguments of [10] show that Γ 2 is discrete, D is a fundamental domain for it, and that these relations present the group. This made a good confidence–building exercise for me, and might do the same for other people. Note that this D is simpler than the Ford domain for 5 discussed in [11], so Γ really is an 2 intermediate example. 5. Completion of the discovery This procrastination had now given me a bigger worry which can be put thus: Why should the Great Lord have performed a unique miracle to make Γ discrete, for no visible reason at all?! The answer is compelling: He didn’t! If Γ is discrete then many other groups have to be discrete, in direct defiance of Shimizu’s lemma, and, since each case of discreteness requires a good reason, there must be general theorems explaining this discreteness. It is a little ironic that this prediction was amply vindicated for (suitable) 3–manifold groups, but, at this writing, the general theorem explaining the discreteness of Γ has not been stated, let alone proved. 10 ROBERT RILEY During a few weeks further procrastination the above considerations compelled me to predict that a knot in S3 is hyperbolic unless it clearly was not. Early in March 1974, I think, I finally went to work on 5 and in a few hours had confirmed my prediction. This completed 2 the essential part of my discovery, and all later cases, such as 7 and 4 several links, were just routine examples at most illustrating matters of secondary importance, such as the symmetries of a knot. In fact, for a while I was confused by the symmetries and thought that a too– rich symmetry group would preclude the hyperbolic structure, but I eventually found my mistake. So by late 1974 I had gotten it right: a knotis hyperbolic unlessits groupcontainsa noncyclicabeliansubgroup which is not peripheral. Making bold sweeping conjectures is unnatural for me, and I didn’t venture to predict anything about arbitrary 3– manifolds. I suppose that I might have predicted which 3–manifolds were hyperbolic had someone pressed me on the issue in conversation, but I was too isolated and unknown for that to happen. The locals at Southampton were rather cool about the whole project, except for David Singerman. He liked it enough to propose that we try to get the Science Research Council (of Great Britain) to support me on a hyperbolic project at Southampton University while I got my Ph.D. and looked for a permanent job. His plan was to time the submission of the proposal so that the referee would be at the summer 1975 conference on Kleinian groups at Cambridge where I would publicize hyperbolic structure. Whether or not the plan worked, the Kleinian groupies liked my examples, especially because these examples pointed up the importance of their own work. The SRC did fund the project generously, ultimately for four years 1976–1979. The first two years of the project were devoted to the development of the system PNCRE [10], a file of Fortran subroutines to compute with explicit subgroups of SL (C). PNCRE was not easy to develop and 2 its first output came early in 1977. Meanwhile, about March 1976, a colleague gave me a preprint of Thurston’s lecture [13] on foliations of surfaces. This was the first I heard of him, and I recall that on reading it I became certain that he and I would never share any common mathematical interest. In late June 1976 a friend drove me up to the University of Warwick to hear a lecture by J. Milnor on topics like Sarkovskii’s theorem. Directly he was finished I very nervously (read: scared stiff) introduced myself to him and told him about examples of hyperbolic knots/links. He was interested, and asked a number of direct questions, so that in a minute he understood the status of my project (examples only). I did not guess that he already knew something about