|
Showing 1 - 6 of
6 matches in All Departments
Over the last fifteen years, the face of knot theory has changed
due to various new theories and invariants coming from physics,
topology, combinatorics and alge-bra. It suffices to mention the
great progress in knot homology theory (Khovanov homology and
Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give
rise to strong invariants of knots and 3-manifolds, in particular,
many new unknot detectors. New to this Edition is a discussion of
Heegaard-Floer homology theory and A-polynomial of classical links,
as well as updates throughout the text. Knot Theory, Second Edition
is notable not only for its expert presentation of knot theory's
state of the art but also for its accessibility. It is valuable as
a profes-sional reference and will serve equally well as a text for
a course on knot theory.
Over the last fifteen years, the face of knot theory has changed
due to various new theories and invariants coming from physics,
topology, combinatorics and alge-bra. It suffices to mention the
great progress in knot homology theory (Khovanov homology and
Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give
rise to strong invariants of knots and 3-manifolds, in particular,
many new unknot detectors. New to this Edition is a discussion of
Heegaard-Floer homology theory and A-polynomial of classical links,
as well as updates throughout the text. Knot Theory, Second Edition
is notable not only for its expert presentation of knot theory's
state of the art but also for its accessibility. It is valuable as
a profes-sional reference and will serve equally well as a text for
a course on knot theory.
The book is the first systematic research completely devoted to a
comprehensive study of virtual knots and classical knots as its
integral part. The book is self-contained and contains up-to-date
exposition of the key aspects of virtual (and classical) knot
theory.Virtual knots were discovered by Louis Kauffman in 1996.
When virtual knot theory arose, it became clear that classical knot
theory was a small integral part of a larger theory, and studying
properties of virtual knots helped one understand better some
aspects of classical knot theory and encouraged the study of
further problems. Virtual knot theory finds its applications in
classical knot theory. Virtual knot theory occupies an intermediate
position between the theory of knots in arbitrary three-manifold
and classical knot theory.In this book we present the latest
achievements in virtual knot theory including Khovanov homology
theory and parity theory due to V O Manturov and graph-link theory
due to both authors. By means of parity, one can construct
functorial mappings from knots to knots, filtrations on the space
of knots, refine many invariants and prove minimality of many
series of knot diagrams.Graph-links can be treated as "diagramless
knot theory": such "links" have crossings, but they do not have
arcs connecting these crossings. It turns out, however, that to
graph-links one can extend many methods of classical and virtual
knot theories, in particular, the Khovanov homology and the parity
theory.
This book consists of a selection of articles devoted to new ideas
and developments in low dimensional topology. Low dimensions refer
to dimensions three and four for the topology of manifolds and
their submanifolds. Thus we have papers related to both manifolds
and to knotted submanifolds of dimension one in three (classical
knot theory) and two in four (surfaces in four dimensional spaces).
Some of the work involves virtual knot theory where the knots are
abstractions of classical knots but can be represented by knots
embedded in surfaces. This leads both to new interactions with
classical topology and to new interactions with essential
combinatorics.
This is the second of a three-volume set collecting the original
and now-classic works in topology written during the 1950s1960s.
The original methods and constructions from these works are
properly documented for the first time in this book. No existing
book covers the beautiful ensemble of methods created in topology
starting from approximately 1950, that is, from Serre's celebrated
"singular homologies of fiber spaces."
This book contains an in-depth overview of the current state of the
recently emerged and rapidly growing theory of Gnk groups,
picture-valued invariants, and braids for arbitrary manifolds.
Equivalence relations arising in low-dimensional topology and
combinatorial group theory inevitably lead to the study of
invariants, and good invariants should be strong and apparent. An
interesting case of such invariants is picture-valued invariants,
whose values are not algebraic objects, but geometrical
constructions, like graphs or polyhedra.In 2015, V O Manturov
defined a two-parametric family of groups Gnk and formulated the
following principle: if dynamical systems describing a motion of n
particles possess a nice codimension 1 property governed by exactly
k particles then these dynamical systems possess topological
invariants valued in Gnk.The book is devoted to various
realisations and generalisations of this principle in the broad
sense. The groups Gnk have many epimorphisms onto free products of
cyclic groups; hence, invariants constructed from them are powerful
enough and easy to compare. However, this construction does not
work when we try to deal with points on a 2-surface, since there
may be infinitely many geodesics passing through two points. That
leads to the notion of another family of groups - nk, which give
rise to braids on arbitrary manifolds yielding invariants of
arbitrary manifolds.
|
You may like...
Loot
Nadine Gordimer
Paperback
(2)
R398
R330
Discovery Miles 3 300
Hampstead
Diane Keaton, Brendan Gleeson, …
DVD
R66
Discovery Miles 660
|