Algebraic Invariants of Links of Codimension Two PDF Download

Author: Nobuyuki Albert Sato
Publisher:
ISBN:
Category : Algebraic number theory
Languages : en
Pages : 59

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Author: Jonathan Hillman
Publisher: World Scientific
ISBN: 9814407402
Category : Mathematics
Languages : en
Pages : 370

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Book Description
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.

Author: Jonathan Arthur Hillman
Publisher: World Scientific
ISBN: 9814407380
Category : Mathematics
Languages : en
Pages : 370

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Book Description
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.

Author: Jonathan Arthur Hillman
Publisher: World Scientific
ISBN: 9814407399
Category : Mathematics
Languages : en
Pages : 370

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Book Description
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 OCo 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.

Author: Erica Flapan
Publisher: American Mathematical Soc.
ISBN: 1470428474
Category : Mathematics
Languages : en
Pages : 202

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Book Description
This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24–25, 2015, at California State University, Fullerton, CA. Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves. The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in and other 3-manifolds.

Author: Desmond Sheiham
Publisher: American Mathematical Soc.
ISBN: 9780821865064
Category : Mathematics
Languages : en
Pages : 132

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An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S^n \subset S^{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. An $F_\mu$-link is a boundary link together with a cobordism class of such spanning manifolds. The $F_\mu$-link cobordism group $C_n(F_\mu)$ is known to be trivial when $n$ is even but not finitely generated when $n$ is odd. Our main result is an algorithm to decide whether two odd-dimensional $F_\mu$-links represent the same cobordism class in $C_{2q-1}(F_\mu)$ assuming $q>1$. We proceed to compute the isomorphism class of $C_{2q-1}(F_\mu)$, generalizing Levine's computation of the knot cobordism group $C_{2q-1}(F_1)$. Our starting point is the algebraic formulation of Levine, Ko and Mio who identify $C_{2q-1}(F_\mu)$ with a surgery obstruction group, the Witt group $G^{(-1)^q,\mu}(\mathbb{Z})$ of $\mu$-component Seifert matrices. We obtain a complete set of torsion-free invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to `algebraically integral' simple self-dual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties. To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing $G^{(-1)^q,\mu}(\mathbb{Q})$ as an infinite direct sum of Witt groups of finite-dimensional division $\mathbb{Q}$-algebras with involution. The Witt group of every such algebra appears as a summand infinitely often. The theory of symmetric and hermitian forms over these division algebras is well-developed. There are five classes of algebras to be considered; complete Witt invariants are available for four classes, those for which the local-global principle applies. An algebra in the fifth class, namely a quaternion algebra with non-standard involution, requires an additional Witt invariant which is defined if all the local invariants vanish.

Author: Andrew Ranicki
Publisher: Springer Science & Business Media
ISBN: 3662120119
Category : Mathematics
Languages : en
Pages : 669

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Bringing together many results previously scattered throughout the research literature into a single framework, this work concentrates on the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory.

Author: Desmond Sheiham
Publisher: American Mathematical Soc.
ISBN: 0821833405
Category : Mathematics
Languages : en
Pages : 128

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Book Description
An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S DEGREESn \subset S DEGREES{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. This title proceeds to compute the isomorphism class of $C_{

Author: Akio Kawauchi
Publisher: Birkhäuser
ISBN: 3034892276
Category : Mathematics
Languages : en
Pages : 431

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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: Tim D. Cochran
Publisher: American Mathematical Soc.
ISBN: 9780821824894
Category : Mathematics
Languages : en
Pages : 102

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Book Description
We investigate higher-order cohomology operations (Massey products) on complements of links of circles in [italic]S3. These are known to be essentially equivalent to the [lowercase Greek]Mu [with macron]-invariants of John Milnor, which detect whether or not the longitudes of the link lie in the [italic]n[superscript]th term of the lower central series of the fundamental group of the link compliment. We define a geometric "derivative" on the set of all links and use this to define higher-order linking numbers which are shown to be "pieces" of Massey products.