Overview
- Shows how the quandle has been evaluated in relation to mathematics or topology while the quandle was often considered to be something combinatorial
- Constitutes a guide on quandles at a time when few surveys of quandles and few topological books on quandles exist
- Emphasizes the geometric advantages of quandles at a high level mathematically while the quandle is used as an algebraic method in many books
- Includes supplementary material: sn.pub/extras
Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)
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Table of contents (8 chapters)
Keywords
About this book
More precisely, this work emphasizes the fresh perspective that quandle theory can be useful for the study of low-dimensional topology (e.g., knot theory) and relative objects with symmetry. The direction of research is summarized as “We shall thoroughly (re)interpret the previous studies of relative symmetry in terms of the quandle”. The perspectives contained herein can be summarized by the following topics. The first is on relative objects G/H, where G and H are groups, e.g., polyhedrons, reflection, and symmetric spaces. Next, central extensions of groups are discussed, e.g., spin structures, K2 groups,and some geometric anomalies. The third topic is a method to study relative information on a 3-dimensional manifold with a boundary, e.g., knot theory, relative cup products, and relative group cohomology.
For applications in topology, it is shown that from the perspective that some existing results in topology can be recovered from some quandles, a method is provided to diagrammatically compute some “relative homology”. (Such classes since have been considered to be uncomputable and speculative). Furthermore, the book provides a perspective that unifies some previous studies of quandles.
The former part of the book explains motivations for studying quandles and discusses basic properties of quandles. The latter focuses on low-dimensional topology or knot theory. Finally, problems and possibilities for future developments of quandle theory are posed.
Reviews
“This short monograph is packed a great deal of interesting mathematics. The book concerns quandles, algebraic structures with axioms related to the Reidemeister moves, and their applications to knot and link invariants. … An interesting connection with Chern-Simons classes rounds out this fun and interesting monograph.” (Sam Nelson, zbMATH 1411.57001, 2019)
“This book aims to be a crash course in quandle theory, and it achieves this goal. … the present book is unique in its emphasis on the homotopy theory of the rack space and the relation to group cohomology. … There is still much worth exploring in this direction, and the author does an excellent job bringing the reader to the front of current research.” (Markus Szymik, Mathematical Reviews, July, 2018)Authors and Affiliations
Bibliographic Information
Book Title: Quandles and Topological Pairs
Book Subtitle: Symmetry, Knots, and Cohomology
Authors: Takefumi Nosaka
Series Title: SpringerBriefs in Mathematics
DOI: https://doi.org/10.1007/978-981-10-6793-8
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s) 2017
Softcover ISBN: 978-981-10-6792-1Published: 28 November 2017
eBook ISBN: 978-981-10-6793-8Published: 20 November 2017
Series ISSN: 2191-8198
Series E-ISSN: 2191-8201
Edition Number: 1
Number of Pages: IX, 136
Number of Illustrations: 14 b/w illustrations, 11 illustrations in colour
Topics: Topology, Group Theory and Generalizations, K-Theory