Quandles and Topological Pairs: Symmetry, Knots, and Cohomology (SpringerBriefs in Mathematics)

This book surveys quandle theory, starting from basic motivations and going on to introduce recent developments of quandles with topological applications and related topics. The book is written from topological aspects, but it illustrates how esteemed quandle theory is in mathematics, and it constitutes a crash course for studying quandles.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, __K__2 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. Preface 5 Contents 6 Notational Conventions 8 1 Introduction 9 2 Basics of Quandles 12 2.1 Definitions and Examples of Quandles 12 2.2 Characterization Theorem of Quandles from Groups 16 2.3 Quandles and Centrally Extended Groups 19 2.4 Link Quandles and Their Properties 21 2.5 Appendix: Historical and Topological Comments on Quandles 23 3 X-Colorings of Links 26 3.1 Definitions and Examples 26 3.2 Characterization of X-colorings 30 3.2.1 Example 1; Parabolic SL2-Representations 32 3.2.2 Example 2; Core Quandle and Double Branched Covering 34 3.2.3 Example 3; First Cohomology with Local Coefficients 35 3.3 Appendix: Colorings and Braid Actions 37 4 Some of Quandle Cocycle Invariants of Links 40 4.1 Origin: The Homotopy Invariant from the Rack Space 40 4.2 Shadow Cocycle Invariant Associated with X-sets 43 4.3 Cocycle Invariant with Non-Abelian Coefficients 46 4.4 Binary and Trinary Cocycle Invariants 48 5 Topology of the Rack Space and the 2-Cocycle Invariant 52 5.1 Rack and Quandle Homology with Properties 52 5.2 The Rack Space and its Properties 55 5.3 Topology of the Non-Abelian Cocycle Invariants 59 5.4 Examples; Second Quandle Homology and 2-Cocycles 62 6 Topology on the Quandle Homotopy Invariant 65 6.1 Monoid Structure on the Universal Covering 65 6.2 On the Classifying Map of the Rack Space 67 6.3 Rack Spaces of the Knot Quandles 70 6.4 Topological Interpretation of the Homotopy Invariant 71 6.5 Application; Computation of Third Quandle Homology 74 7 Relative Group Homology 77 7.1 Review; Group Homology 77 7.2 Relative Group Homology as a Mapping Cone 80 7.3 Hochschild Relative Group Homology 82 7.4 Examples of (Relative) Group (Co)cycles 85 7.4.1 Example 1; Group 3-Cocycles of Cyclic Groups 85 7.4.2 Example 2; the Chern-Simons 3-Class 86 7.4.3 Example 3; Presented Groups and Knot Groups 87 8 Inoue--Kabaya Chain Map 0 8.1 Original Definition of Inoue--Kabaya Chain Map 89 8.2 Guideline to Applications and Fundamental Classes 91 8.3 Relation to the Chern-Simons Invariants of Hyperbolic Links 92 8.4 Lifts of Inoue--Kabaya Chain Map 95 8.4.1 For Some Quandle Operations on Groups 95 8.4.2 Recovery of All Mochizuki's 3-Cocycles 98 8.4.3 Quandle Cocycles from Invariant Theory 100 8.4.4 Binary Cocycle Invariants Versus Cohomology Pairings 103 Appendix A Notation and Basic Facts in Knot Theory 105 A.1 Cyclic Branched Coverings, and Hyperbolic Links 109 Appendix B Automorphism Groups from Quandles 112 B.1 Calculations of Inner Automorphism Groups 112 B.2 Quotients and Presentations on Quandles 115 B.3 Examples; Alexander and Core Quandles 117 B.4 Alexanderizations of Quandles 120 Appendix C Small Quandles, and Some Quandle Homology 123 C.1 Classification of Small Quandles 123 C.2 Known Results on Quandle Homology 125 C.3 Some Cocycles of Alexander Quandles 127 References 130 Index 137 This book surveys quandle theory, starting from basic motivations and going on to introduce recent developments of quandles with topological applications and related topics. The book is written from topological aspects, but it illustrates how esteemed quandle theory is in mathematics, and it constitutes a crash course for studying quandles. 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.-- Provided by publisher Annotation This book surveys quandle theory, starting from basic motivations and going on to introduce recent developments of quandles with topological applications and related topics. The book is written from topological aspects, but it illustrates how esteemed quandle theory is in mathematics, and it constitutes a crash course for studying quandles. 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 This book surveys quandle theory, starting from basic motivations and going on to introduce recent developments of quandles with topological applications and related topics. The book is written from topological aspects, but it illustrates how esteemed quandle theory is in mathematics, and it constitutes a crash course for studying quandles.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, K 2 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. This book surveys quandle theory, starting from basic motivations and going on to introduce recent developments of quandles with topological applications and related topics. The book is written from topological aspects, but it illustrates how esteemed quandle theory is in mathematics, and it constitutes a crash course for studying quandles. 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 zWe shall thoroughly (re)interpret the previous studies of relative symmetry in terms of the quandley. 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 zrelative homologyy. (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 Front Matter ....Pages i-ix Introduction (Takefumi Nosaka)....Pages 1-3 Basics of Quandles (Takefumi Nosaka)....Pages 5-18 X-Colorings of Links (Takefumi Nosaka)....Pages 19-32 Some of Quandle Cocycle Invariants of Links (Takefumi Nosaka)....Pages 33-44 Topology of the Rack Space and the 2-Cocycle Invariant (Takefumi Nosaka)....Pages 45-57 Topology on the Quandle Homotopy Invariant (Takefumi Nosaka)....Pages 59-70 Relative Group Homology (Takefumi Nosaka)....Pages 71-82 Inoue–Kabaya Chain Map (Takefumi Nosaka)....Pages 83-98 Back Matter ....Pages 99-136