W. Menasco – Handbook of Knot Theory

Автор: W. Menasco
Название книги: Handbook of Knot Theory
Формат: PDF
Жанр: Математика
Страницы: 503
Качество: Изначально компьютерное, E-book

This book is a survey of current topics in the mathematical theory of knots. For a mathematician, a knot is a closed loop in 3-dimensional space: imagine knotting an extension cord and then closing it up by inserting its plug into its outlet. Knot theory is of central importance in pure and applied mathematics, as it stands at a crossroads of topology, combinatorics, algebra, mathematical physics and biochemistry.

* Survey of mathematical knot theory
* Articles by leading world authorities
* Clear exposition, not over-technical
* Accessible to readers with undergraduate background in mathematics

Accessible to the layman, with immense raw geometric appeal, yet in close touch with
some elevated disciplines, knot theory holds a special place in mathematics. From its
humble beginnings as an empirical science in the mid-19th century, the subject has
undergone an extraordinary evolution, greatly accelerated over the last 25 years. One of
the first serious researchers in knot theory, the notable Scottish physicist Peter Guthrie
Tait, began his celebrated tabulation of knots of small crossing-number with the hope of
elucidating the Kelvin vortex theory of atoms, but soon became entranced with the subject
in its own right. He observed, perceptively, that he could not ever prove that two knots
were inequivalent; indeed, that had to wait until the advent of Poincare´’s Fundamental
Group in the early 1900s. Knot theorists now have a formidable arsenal of techniques for
distinguishing knot and link types, but such considerations form only a narrow part of
contemporary knot theory. Knots have found themselves involved with almost every
major advance in low-dimensional topology, be it Papakyriakopoulos’s proof of Dehn’s
Lemma in 1957 (whose original aim was to show that the group of a non-trivial knot could
not be isomorphic to the integers), or Riley’s discovery in the 1970s of a hyperbolic
structure on the complement of the figure-eight knot and Thurston’s subsequent profound
work on geometric structures on 3-manifolds. Of equal importance are interactions with
piecewise-linear 3-dimensional topology, with braids, with contact structures on
manifolds, and with 4-dimensional topology, both classical and gauge-theoretical. It
would be audacious to attempt even a summary of the full ramifications of knot theory, but
mention must be made of Jones’s astonishing discovery in 1984 of an entirely new breed
of polynomial invariants, which strengthened knot theory’s connection with the theory of
braids and forged a completely new relationship with the mathematics of quantum field
theory.
In this volume, we present a collection of survey articles by leading experts on a crosssection
of present-day knot theory. For us, reading these articles and witnessing the vast
range of knowledge contained therein was an inspiring and humbling experience.
We are grateful for this opportunity to thank the contributors for their sheer hard work
and their willingness to share this knowledge. We would also like to thank the staff of
Elsevier for their copious help with this project.
William Menasco