**Author**: Samuel J. Lomonaco

**Publisher:** American Mathematical Soc.

**ISBN:**

**Category:** Mathematics

**Page:** 346

**View:** 617

This volume arose from a special session on Low Dimensional Topology organized and conducted by Dr. Lomonaco at the American Mathematical Society meeting held in San Francisco, California, January 7-11, 1981.

This book highlights the latest advances in algebraic topology, from homotopy theory, braid groups, configuration spaces and toric topology, to transformation groups and the adjoining area of knot theory. It consists of well-written original research papers and survey articles by subject experts, most of which were presented at the “7th East Asian Conference on Algebraic Topology” held at the Indian Institute of Science Education and Research (IISER), Mohali, Punjab, India, from December 1 to 6, 2017. Algebraic topology is a broad area of mathematics that has seen enormous developments over the past decade, and as such this book is a valuable resource for graduate students and researchers working in the field.

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters — twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters. Sample Chapter(s) Chapter 1: Links (205 KB) Contents:Abelian Covers:LinksHomology and Duality in CoversDeterminantal InvariantsThe Maximal Abelian CoverSublinks and Other Abelian CoversTwisted Polynomial InvariantsApplications: Special Cases and Symmetries:Knot ModulesLinks with Two ComponentsSymmetriesSingularities of Plane Algebraic CurvesFree Covers, Nilpotent Quotients and Completion:Free CoversNilpotent QuotientsAlgebraic ClosureDisc Links Readership: Graduate students and academics in geometry and topology. Keywords:Algebraic Invariant;Links;Abelian Cover;Twisted Polynomial Invariant;Knot ModuleReviews: “The main change to the book is the addition of two chapters. These are timely and welcome additions to an already useful and comprehensive book.” Mathematical Reviews Reviews of the First Edition: “Jonathan Hillman has successfully filled an unfortunate gap in the literature of low-dimensional topology … With this up-to-date book we have a reference covering a range of topics that until now were available only in their original sources … Hillman has done an excellent job of referencing his book, both within the text and in the bibliography, with several hundred references included.” Mathematical Reviews “Algebraic Invariants of Links is masterful, offering a survey of work, much of which has not been summarized elsewhere. It is an essential reference for those interested in link theory … it is unique and valuable.” Bulletin of the American Mathematical Society “The author, who is one of the major experts on the topic, must be surely congratulated for this attractive book, written in a careful, very precise and quite readable style. It serves as an excellent self-contained and up-to-date monograph on the algebraic invariants of links … I strongly recommend this beautiful book to anyone interested in the algebraic theory of links and its applications.” Mathematics Abstracts

This volume presents the proceedings from the conference on low dimensional topology held at the University of Madeira (Portugal). The event was attended by leading scientists in the field from the U.S., Asia, and Europe. The book has two main parts. The first is devoted to the Poincare conjecture, characterizations of $PL$-manifolds, covering quadratic forms of links and to categories in low dimensional topology that appear in connection with conformal and quantum field theory. The second part of the volume covers topological quantum field theory and polynomial invariants for rational homology 3-spheres, derived from the quantum $SU(2)$-invariants associated with the first cohomology class modulo two, knot theory, and braid groups. This collection reflects development and progress in the field and presents interesting and new results.

Volume 2 is divided into three parts: the first 'Surfaces' contains an article by Thurston on earthquakes and by Penner on traintracks. The second part is entitled 'Knots and 3-Manifolds' and the final part 'Kleinian Groups'.

This volume presents 19 refereed articles written by participants in the Singapore International Symposium in Topology and Geometry (SISTAG), held July 2-6, 2001, at the National University of Singapore. Rather than being a simple snapshot of the meeting in the form of a proceedings, it serves as a commemorative volume consisting of papers selected to show the diversity and depth of the mathematics presented at SISTAG. The book contains articles on low-dimensional topology, algebraic, differential and symplectic geometry, and algebraic topology. While papers reflect the focus of the conference, many documents written after SISTAG and included in this volume represent the most up-to-date thinking in the fields of topology and geometry. While representation from Pacific Rim countries is strong, the list of contributors is international in scope and includes many recognized experts. This volume is of interest to graduate students and mathematicians working in the fields of algebraic, differential and symplectic geometry, algebraic, geometric and low-dimensional topology, and mathematical physics.