There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields.
Author: Lee Paul Neuwirth
Publisher: Princeton University Press
There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields. These papers are dedicated to the memory of Ralph H. Fox, one of the world's leading topologists, by colleagues, former students, and friends. In knot theory, papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten. Of these several are devoted to the study of branched covering spaces over knots and links, while others utilize the braid groups of Artin. Cossey and Smythe, Stallings, and Strasser address themselves to group theory. In his contribution Stallings describes the calculation of the groups In/In+1 where I is the augmentation ideal in a group ring RG. As a consequence, one has for each k an example of a k-generator l-relator group with no free homomorphs. In the third part, papers by Birman, Cappell, Milnor, Montesinos, Papakyriakopoulos, and Shalen comprise the treatment of 3-manifolds. Milnor gives, besides important new results, an exposition of certain aspects of our current knowledge regarding the 3- dimensional Brieskorn manifolds.
J. W. Milnor, On the total curvature of knots, Ann. of Math. (2) 52 (1950), 248—257. , On the 3-dimensional Brieskom manifolds M (p, q, r), Knots, Groups and 3manifolds (L. P. Neuwirth, ed.), Ann. of Math. Studies, vol. 84 ...
Author: V. V. Prasolov
Publisher: American Mathematical Soc.
This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of 3-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones-Witten invariants are constructed via the rigorous skein algebra approach (mainly due to the Saint Petersburg school). Unlike several recent monographs, where all of these invariants are introduced by using the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites are minimal in this book. Numerous figures and problems make it suitable as a course text and for self-study.
S. Kojima, Isometry transformations of hyperbolic 3-manifolds, Topology Appl. 29 (1988), 297–307. ... Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. Math. Studies, vol. 84, Princeton Univ.
Author: Arnaud Deruelle
Publisher: American Mathematical Soc.
Category: Complex manifolds
The authors propose a new approach in studying Dehn surgeries on knots in the $3$-sphere $S^3$ yielding Seifert fiber spaces. The basic idea is finding relationships among such surgeries. To describe relationships and get a global picture of Seifert surgeries, they introduce ``seiferters'' and the Seifert Surgery Network, a $1$-dimensional complex whose vertices correspond to Seifert surgeries. A seiferter for a Seifert surgery on a knot $K$ is a trivial knot in $S^3$ disjoint from $K$ that becomes a fiber in the resulting Seifert fiber space. Twisting $K$ along its seiferter or an annulus cobounded by a pair of its seiferters yields another knot admitting a Seifert surgery. Edges of the network correspond to such twistings. A path in the network from one Seifert surgery to another explains how the former Seifert surgery is obtained from the latter after a sequence of twistings along seiferters and/or annuli cobounded by pairs of seiferters. The authors find explicit paths from various known Seifert surgeries to those on torus knots, the most basic Seifert surgeries. The authors classify seiferters and obtain some fundamental results on the structure of the Seifert Surgery Network. From the networking viewpoint, they find an infinite family of Seifert surgeries on hyperbolic knots which cannot be embedded in a genus two Heegaard surface of $S^3$.
Topology 2 , 275-280 . Problem : 1.61 . ( 1064 ) Trotter , H. F. ( 1975 ) . Some knots spanned by more than one unknotted surface of minimal genus . In L. P. Neuwirth ( Ed . ) , Knots , Groups , and 3 - Manifolds , Volume 84 of Ann . of ...
Author: William Hilal Kazez
Publisher: American Mathematical Soc.
This is Part 2 of a two-part volume reflecting the proceedings of the 1993 Georgia International Topology Conference held at the University of Georgia during the month of August. The texts include research and expository articles and problem sets. The conference covered a wide variety of topics in geometric topology. Features: Kirby's problem list, which contains a thorough description of the progress made on each of the problems and includes a very complete bibliography, makes the work useful for specialists and non-specialists who want to learn about the progress made in many areas of topology. This list may serve as a reference work for decades to come. Gabai's problem list, which focuses on foliations and laminations of 3-manifolds, collects for the first time in one paper definitions, results, and problems that may serve as a defining source in the subject area.
J.P. Mayberry, K. Murasugi, Torsion groups of abelian coverings of links. Trans. Am. Math. Soc. 271, 143–173 (1982) 199. C.T. McMullen, Renormalization and 3-Manifolds Which Fiber over the Circle. Annals of Mathematics Studies, vol.
Author: Ken’ichi Ohshika
Publisher: Springer Nature
This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups, foliations, mapping class groups, Teichmüller theory, anti-de Sitter geometry, and co-Minkowski geometry. The book is addressed to researchers and students who want to learn about Thurston’s wide-ranging mathematical ideas and their impact. At the same time, it is a tribute to Thurston, one of the greatest geometers of all time, whose work extended over many fields in mathematics and who had a unique way of perceiving forms and patterns, and of communicating and writing mathematics.
Annals of Mathematics Studies, No. 72, Princeton University Press (1971).  J. W. Milnor, On the Brieskorn (p, q, r)-manifolds, In Knots, Groups and 3-Manifolds, pp. 175–225. Annals of Mathematics Studies, Vol. 84.
Author: Bullett Shaun
Publisher: World Scientific
This book leads readers from a basic foundation to an advanced level understanding of geometry in advanced pure mathematics. Chapter by chapter, readers will be led from a foundation level understanding to advanced level understanding. This is the perfect text for graduate or PhD mathematical-science students looking for support in algebraic geometry, geometric group theory, modular group, holomorphic dynamics and hyperbolic geometry, syzygies and minimal resolutions, and minimal surfaces. Geometry in Advanced Pure Mathematics is the fourth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Editor the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.
[Sha90a] Shalev, A. Dimension subgroups, nilpotency indices, and the number of generators of ideals in p-group algebras. J. Algebra, 129:412–438, ... [Sta75] Stallings, J. Knots, Groups and 3-Manifolds, volume 84 of Annals of Math.
Author: Roman Mikhailov
Publisher: Springer Science & Business Media
A fundamental object of study in group theory is the lower central series of groups. Understanding its relationship with the dimension series is a challenging task. This monograph presents an exposition of different methods for investigating this relationship.
R. Azencott and E. Wilson, Homogeneous manifolds with negative curvature, Part I, Trans. ... 3-dimensional Brieskorn manifolds M(p, q, r), in "Knots, Groups, and 3-Manifolds," pp. 175-225, Ann. Math. Studies, Vol. 84, Princeton Univ.
Author: John Willard Milnor
Publisher: American Mathematical Soc.
This volume contains papers on geometry of one of the best modern geometers and topologists, John Milnor. This book covers a wide variety of topics and includes several previously unpublished works. It is delightful reading for any mathematician with an interest in geometry and topology and for any person with an interest in mathematics. (A number of papers in the collection, intended for a general mathematical audience, have been published in the American Mathematical Monthly.) Each paper is accompanied by the author's comments on further development of the subject. The volume contains twenty-one papers and is partitioned into three parts: Differential geometry and curvature, Algebraic geometry and topology, and Euclidean and non-Euclidean geometry. Although some of the papers were written quite a while ago, they appear more modern than many of today's publications. Milnor's excellent, clear, and laconic style makes the book a real treat. This volume is highly recommended to a broad mathematical audience, and, in particular, to young mathematicians who will certainly benefit from their acquaintance with Milnor's mode of thinking and writing.Information for our distributors: A publication of Publish or Perish, Inc.
 Covering properties of small volume hyperbolic 3-manifolds, J . Knot Theory Ramification 7 (1998), 381-392. ... equal volumes but different Chem—Simons invariants, Lowdimensional topology and Kleinian groups (Coventry/Durharn 1984), ...
Author: Boris N. Apanasov
Publisher: Walter de Gruyter
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany
357–370  J. Milnor, "On the 3-dimensional Brieskorn Manifolds", pp. 175–225 in Knots, Groups and 3-Manifolds, Annals of Math. Studies #84, edited by L. P. Neuwirth, Princeton Univ. Press 1975  J. Minkus, (a) Abstract #77T-A9, ...