Invariants of links and 3-manifolds from graph configurations / Christine Lescop.

Format:

Book

Author/Creator:

Lescop, Christine, 1966- author.

Series:

EMS monographs in mathematics

Language:

English

Subjects (All):

Knot theory.

Three-manifolds (Topology).

Graph theory.

Physical Description:

xv, 571 pages ; 24 cm

Place of Publication:

Berlin, Germany : European Mathematical Society, 2024.

Summary:

"This self-contained book explains how to count graph configurations to obtain topological invariants for 3-manifolds and links in these 3-manifolds, and it investigates the properties of the obtained invariants. The simplest of these invariants is the linking number of two disjoint knots in the ambient space described in the beginning of the book as the degree of a Gauss map. Mysterious knot invariants called "quantum invariants" were introduced in the mid-1980s, starting with the Jones polynomial. Witten explained how to obtain many of them from the perturbative expansion of the Chern-Simons theory. His physicist viewpoint led Kontsevich to a configuration-counting definition of topological invariants for the closed 3-manifolds where knots bound oriented compact surfaces. The book's first part shows in what sense an invariant previously defined by Casson for these manifolds counts embeddings of the theta graph. The second and third parts describe a configuration-counting invariant \mathcal Z generalizing the above invariants. The fourth part shows the universality of \mathcal Z with respect to some theories of finite-type invariants. The most sophisticated presented generalization of \mathcal Z applies to small pieces of links in 3-manifolds called tangles. Its functorial properties and its behavior under cabling are used to describe the properties of \mathcal Z. The book is written for graduate students and more advanced researchers interested in low-dimensional topology and knot theory."--Back cover

Contents:

I. Introduction

Introductions

More on manifolds and the linking number

Propagators

The Theta invariant

Parallelizations of 3-manifolds and Pontrjagin classes

Part II The general invariants

Introduction to finite type invariants and Jacobi diagrams

First definitions of Z

Compactifications of configuration spaces

Dependence on the propagating forms

First properties of Z and anomalies

Rationality

Part III Functoriality

A first introduction to the functor Zf

More on the functor Zf

Invariance of Zf for long tangles

The invariant Z as a holonomy for braids

Discretizable variants of Zf and extensions to q-tangles

Justifying the properties of Zf

Part IV Universality

The main universality statements and their corollaries

More flexible definitions of Z using pseudo-parallelizations

Simultaneous normalization of propagating forms

Much more flexible definitions of Z.

Notes:

Includes bibliographical references and index.

ISBN:

3985470820

9783985470822

OCLC:

1483640393