New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots Francis Bonahon & Laurence C. Siebenmann Department of Mathematics Math´ematique, Bˆatiment 425 University of Southern California Universit´ede Paris-Sud LosAngeles,CA90089-2532,U.S.A. 91405OrsayCedex,France Version: January 26, 2016 Copyright: Francis Bonahon and Laurent Sienbenmann, 1979-2010 Preface to the 30-th year edition Dedicated to Jos´eMontesinos on the occasion of his 65-th birthday We began writing this monograph in 1979. It originally started as an article for the conference proceedings [BroT], but it soon developed into a book project of its own as the size of the manuscript was rapidly increasing. The two articles [BonS1] and [BonS2] both originated as appendices to this book, as did the unpublished part of the preprint [BonS3] that is dedicated to crystallographic groups. As time passed by, we eventually became bogged down in the process, in large part because of the overwhelming number of pictures and diagrams (about 640 in the current version) and because of the rapid changes in technology occurring at the same time. The book started as a camera-ready typescript with ink drawings, but we soon had to move to the new norm of TEX and computer drawn pictures. In particular, the book was used as developing ground for Sweet-TEX[Sie3], a TEX preprocessor for Macintosh computers created by the second author; this had the unfortunate outcome that many files became obsolete when Apple completely redesigned its operating system. The fact that the authors became geographically separated, and eventually drifted towards different scientific interests, did not help. By the late 1980s, we had collected the then-current state of the manuscript into an “official version”, that we made freely available by request. However, very few pictures were inserted in the text, it lacked bibliographic references, and it incorporated very different styles. By all accounts, it was pretty much unreadable to anybody but the authors. The next 20 years saw no change in the state of the manuscript, until recently. On the other hand, this work seems to have been reasonably influential. For instance, a quick Internet search returns about 100 articles and books referring to it (under many different titles!). We cannot help noting this fact with a certain satisfaction, in particular if one takes into account that these references point to an unpublished document that was difficult to access even as a preprint. Most of the theoretical results underpinning this work had appeared in [BonS1] in a very general form, but there clearly was a need for a precise description of their applications to knot theory, and of the algorithmic methods that we had developed. Combined with persistent exhortations from various colleagues and friends, this provided a strong incentive to resume the task of putting the text in a form suitable for publication. What follows is an extensively edited version of the text in its 1988 form. Many events have occurred in low-dimensional topology in the 30 years that have elapsed since we started this project, the most important one being the completion of the geometrisation program for 3–dimensional manifolds and orbifolds. With the corresponding hindsight and perspective, there are many parts that we would probably write in a very different way right now. However, we decided to preserve iii iv PREFACE TO THE 30-TH YEAR EDITION the historical data, and to remain faithful to the original. We limited our editing to locally improving the clarity of the exposition, and we respected some of the stylistic idiosyncrasies of the text. Updates and references to more recent work are indicated in footnotes. Several people have helped us with the technical aspects of the enterprise, in particular in the early years for typing and picture drawing. This includes Valerie Siviter, Bernadette Barbichon, Bernard Thomas and Banwari Lal Sharma. The majority of the pictures of the current version were drawn by Julien Roger. We are very grateful to all of them, as well as to Rob Kirby, Cameron Gordon and Andrew Ranicki for pushing us to prepare this edition. While preparing this revised version, the first author and Julien Roger were partially supported by the grant DMS-0604866 from the U.S. National Science Foundation. A large part of the mathematics in this book can be traced back to a series of lectures that Jos´eMontesinos gave at Orsay in the spring of 1976. In particular, the first author fondly remembers the many meetings that he had with Jos´ein the Jardin du Luxembourg in Paris, where Jos´esuggested as a possible research topic the classification of what we call here Montesinos knots. This became the dissertation [Bon1], and consequently launched the first author’s career as a mathematician (in addition to providing material for several chapters in this monograph). It is a pleasure to dedicate this book to Jos´eMontesinos as an acknowledgement of our debt, and as a way to celebrate his 65-th birthday and his many contributions to 3–dimensional topology. December 2009 Introduction I was led to the consideration of the forms of knots by Sir W. Thom- son’s1 Theory of Vortex Atoms, and consequently the point of view which at least at first I adopted was that of classifying knots by the number of their crossings. The subject is very much more difficult and intricate than at first sight one is inclined to think, and I feel that I have not suc- ceeded in catching the key-note. When that is found, the various results here given will, no doubt, appear in their real connection with one another, perhaps even as immediate consequences of a thoroughly adequate conception of the question. P.J. Tait [Tai1], 1877. A century after the Scottish physicist P.G. Tait wrote these words, it is still a challenging task for the topologists of this day to acquire a theoretical under- standing of knotting adequate to explain even the 630 or so (connected) knots tabulated in the last century by Tait, T.P. Kirkman and C.N. Little [Tai1, Tai2, Tai3, Kirk1, Kirk2, Lit1, Lit2, Lit3, Lit4]. These XIX-th century tabulations listed connected knots of 6 11 crossings excepting the 11 crossing connected knots hav- ing non-alternating projection. Since then, J.H. Conway [Conw] has tabulated the missing (non-alternating) 11 crossing connected knots plus the links of 6 10 cross- ings and very recently A. Caudron [Cau] has listed the 11 crossing links; concerning tabulations, see also [Has1, Has2]. This is the task that we shall attempt here, and accomplish in part. To Thom- son and Tait, our understanding would be disturbing inasmuch as it is more molec- ular than atomic; indeed the first phenomena we explain are described by integrally weighted trees a bit reminiscent of the organic molecule of the paraffin series. A knot K in the 3–sphere S3 = R3 ∪ {∞} is a collection of smoothly and disjointly embedded circles, namely K is a closed 1–submanifold of S3. Note that our terminology is somewhat non-standard, in the sense that we do not require knots to be connected, nor oriented. A link is a knot with > 2 components. Two 3 3 knots (S ,K1), (S ,K2) are isomorphic or “the same” if there is a degree +1 3 3 (namely orientation-preserving) diffeomorphism f : S → S with f(K1) = K2. Since a degree +1 diffeomorphism of S3 is isotopic to the identity [Cer], this is 3 equivalent to the property that K1 can be deformed to K2 in S by a smooth isotopy. In his remarkable heuristic work [Conw] on knot classification, J.H. Conway made much use of 2–spheres smoothly embedded in S3 and cutting K transversely in 4 points. We call these Conway spheres for the knot (S3,K). Figure 0.1 1= Lord Kelvin, or more precisely Baron Kelvin of Largs. See [Tho] v vi INTRODUCTION provides examples of such Conway spheres, obtained by capping off each of the dotted curves by one disk located in front of the sheet of paper and another disk in the back. Figure 0.1. 3 A knot (S ,K) is arborescent if there exists a finite collection F1,..., Fn of disjoint Conway spheres such that, if N is the closure of any component of 3 n S − i=1 Fi, then the pair (N,K ∩ N) takes the simple form of Figure 0.2 after suitableS isotopic deformation in S3. In this figure, N is S3 with the interior of a finite collection of disjoint balls deleted and K ∩ N is the intersection of N with two standard circles. Figure 0.2. For example the Conway spheres dotted in Figure 0.1 reveal that knot to be arborescent. In other words, arborescent knots are those that can be broken up into (almost) trivial pieces by Conway spheres. It was Conway who singled out this class and first studied their projections [Conw]. He found that over 75% of the knots tabulated in the XIX–th century, including nearly 90% of all knots and links of 6 10 crossings, are arborescent; see also [Cau]. These arborescent knots are called algebraic by Conway because, in the frame- work of [Conw], they are obtained from finitely many copies of the 1–crossing tan- gle by applying a sequence of “sum” and “product” operations; see Chapter 14. Unfortunately, the name of algebraic knot is also used to describe the iterated torus knots that occur as links of algebraic complex singularities; see for instance [Bra, Bur1, Bur2, Zar, EisN]. The two families have very little in common, in INTRODUCTION vii fact just the {2,n}{3, 4} and {3, 5} torus knots, as will be proved in Theorem A.8 in the Appendix.