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Author: Jerome Minkus Publisher: American Mathematical Soc. ISBN: 0821822551 Category : Knot theory Languages : en Pages : 75
Book Description
In this paper a family of closed oriented 3 dimensional manifolds {[italic]M[subscript italic]n([italic]k,[italic]h)} is constructed by pasting together pairs of regions on the boundary of a 3 ball. The manifold [italic]M[subscript italic]n([italic]k,[italic]h) is a generalization of the lens space [italic]L([italic]n,1) and is closely related to the 2 bridge knot or link of type ([italic]k,[italic]h). While the work is basically geometrical, examination of [lowercase Greek]Pi1([italic]M[subscript italic]n([italic]k,[italic]h)) leads naturally to the study of "cyclic" presentations of groups. Abelianizing these presentations gives rise to a formula for the Alexander polynomials of 2 bridge knots and to a description of [italic]H1([italic]M[subscript italic]n([italic]k,[italic]h), [italic]Z) by means of circulant matrices whose entries are the coefficients of these polynomials.
Author: Jerome Minkus Publisher: American Mathematical Soc. ISBN: 0821822551 Category : Knot theory Languages : en Pages : 75
Book Description
In this paper a family of closed oriented 3 dimensional manifolds {[italic]M[subscript italic]n([italic]k,[italic]h)} is constructed by pasting together pairs of regions on the boundary of a 3 ball. The manifold [italic]M[subscript italic]n([italic]k,[italic]h) is a generalization of the lens space [italic]L([italic]n,1) and is closely related to the 2 bridge knot or link of type ([italic]k,[italic]h). While the work is basically geometrical, examination of [lowercase Greek]Pi1([italic]M[subscript italic]n([italic]k,[italic]h)) leads naturally to the study of "cyclic" presentations of groups. Abelianizing these presentations gives rise to a formula for the Alexander polynomials of 2 bridge knots and to a description of [italic]H1([italic]M[subscript italic]n([italic]k,[italic]h), [italic]Z) by means of circulant matrices whose entries are the coefficients of these polynomials.
Author: Akio Kawauchi Publisher: Birkhäuser ISBN: 3034892276 Category : Mathematics Languages : en Pages : 431
Book Description
Knot theory is a rapidly developing field of research with many applications, not only for mathematics. The present volume, written by a well-known specialist, gives a complete survey of this theory from its very beginnings to today's most recent research results. An indispensable book for everyone concerned with knot theory.
Author: Dale Rolfsen Publisher: American Mathematical Soc. ISBN: 0821834363 Category : Mathematics Languages : en Pages : 458
Book Description
Rolfsen's beautiful book on knots and links can be read by anyone, from beginner to expert, who wants to learn about knot theory. Beginners find an inviting introduction to the elements of topology, emphasizing the tools needed for understanding knots, the fundamental group and van Kampen's theorem, for example, which are then applied to concrete problems, such as computing knot groups. For experts, Rolfsen explains advanced topics, such as the connections between knot theory and surgery and how they are useful to understanding three-manifolds. Besides providing a guide to understanding knot theory, the book offers 'practical' training. After reading it, you will be able to do many things: compute presentations of knot groups, Alexander polynomials, and other invariants; perform surgery on three-manifolds; and visualize knots and their complements.It is characterized by its hands-on approach and emphasis on a visual, geometric understanding. Rolfsen offers invaluable insight and strikes a perfect balance between giving technical details and offering informal explanations. The illustrations are superb, and a wealth of examples are included. Now back in print by the AMS, the book is still a standard reference in knot theory. It is written in a remarkable style that makes it useful for both beginners and researchers. Particularly noteworthy is the table of knots and links at the end. This volume is an excellent introduction to the topic and is suitable as a textbook for a course in knot theory or 3-manifolds. Other key books of interest on this topic available from the AMS are ""The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes"" and ""The Knot Book.""
Author: Jonathan C. Johnson (Ph. D.) Publisher: ISBN: Category : Languages : en Pages : 268
Book Description
The orderability of a 3-manifold group is closely connected to the topological properties of the manifold. Link groups are always left-orderable. However, there are link groups which are known to be bi-orderable, as well as link groups known not to be bi-orderable. In this dissertation, the bi-orderability of some families of link groups is shown. We show that two-bridge links with Alexander polynomials whose coefficients are coprime are extensions of Z by residually torsion-free nilpotent groups. It follows from a result of Linnell-Rhemtulla-Rolfsen[30] that if a two-bridge link has an Alexander polynomial with coprime coefficients and all real positive roots, then its link group is bi-orderable. In particular, if a two-bridge knot has an Alexander polynomial with all real positive roots, then its knot group is bi-orderable. This result shows that a large family of knots whose cyclic branched covers are known to be L-spaces have bi-orderable knot groups. Additionally, using a technique developed by Mayland, the pretzel knots P(-3, 3, 2r + 1) are shown to have bi-orderable knot groups. Issa-Turner[20] showed that all the cyclic branched covers of these knots are L-spaces. Finally, a family of genus one pretzel knots are shown to have bi-orderable knot groups and double branched covers which are not L-spaces
Author: Young Gheel Baik Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110807491 Category : Mathematics Languages : en Pages : 392
Book Description
The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
Author: R.B. Sher Publisher: Elsevier ISBN: 0080532853 Category : Mathematics Languages : en Pages : 1145
Book Description
Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys written by leading experts and covering active areas of current research. They provide the reader with an up-to-date overview of this flourishing branch of mathematics.
Author: Jerome Minkus
Author: Jerome Minkus
Author: Jerome Minkus
Author: Akio Kawauchi
Author: Dale Rolfsen
Author: Dale Rolfsen
Author: 
Author: David James Schorow
Author: Jonathan C. Johnson (Ph. D.)
Author: Young Gheel Baik
Author: R.B. Sher