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Virtual knots : the state of the art
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Description
Rating
Title
Virtual
knots
: the
state
of the
art
Creator
Manturov, V. O. (Vasilii Olegovich)
Contributors
Iliutko, D. P. (Denis Petrovich), 1979–
World Scientific (Firm)
DescriptionAbstract
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
selfcontained
and
contains
uptodate
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
threemanifold
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
graphlink
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
.
Graphlinks
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
graphlinks
one
can
extend
many
methods
of
classical
and
virtual
knot
theories
, in
particular
, the
Khovanov
homology
and the
parity
theory
.
DescriptionTable Of Contents
1
.
Basic
definitions
and
notions
.
1.1
.
Classical
knots
.
1.2
.
Virtual
knots
.
1.3
.
Selflinking
number

2
.
Virtual
knots
and
threedimensional
topology
.
2.1
.
Introduction
.
2.2
. The
Kuperberg
theorem
.
2.3
.
Genus
of a
virtual
knot
.
2.4
.
Recognition
of
virtual
links

3
.
Quandles
(distributive
groupoids)
in
virtual
knot
theory
.
3.1
.
Introduction
.
3.2
.
Quandles
and their
generalizations
.
3.3
.
Long
virtual
knots
.
3.4
.
Virtual
knots
and
infinitedimensional
Lie
algebras
.
3.5
.
Hierarchy
of
virtual
knots

4
. The
Jones–Kauffman
polynomial
:
atoms
.
4.1
.
Introduction
.
4.2
.
Basic
definitions
.
4.3
. The
polynomial
[symbol]
:
minimality
problems
.
4.4
.
Rigid
virtual
knots
.
4.5
.
Minimal
diagrams
of
long
virtual
knots

5
.
Khovanov
homology
.
5.1
.
Introduction
.
5.2
.
Basic
constructions
: the
Jones
polynomial
[symbol]
.
5.3
.
Khovanov
homology
with
[symbol]coefficients
.
5.4
.
Khovanov
homology
of
double
knots
.
5.5
.
Khovanov
homology
and
atoms
.
5.6
.
Khovanov
homology
and
parity
.
5.7
.
Khovanov
homology
for
virtual
links
.
5.8
.
Spanning
tree
for
Khovanov
complex
.
5.9
. The
Khovanov
polynomial
and
Frobenius
extensions
.
5.10
.
Minimal
diagrams
of
links

6
.
Virtual
braids
.
6.1
.
Introduction
.
6.2
.
Definitions
of
virtual
braids
.
6.3
.
Virtual
braids
and
virtual
knots
.
6.4
. The
Kauffman
bracket
polynomial
for
braids
.
6.5
.
Invariants
of
virtual
braids

6
.
Vassiliev's
invariants
and
framed
graphs
.
7.1
.
Introduction
.
7.2
. The
Vassiliev
invariants
of
classical
knots
and
Jinvariants
of
curves
.
7.3
. The
Goussarov–Polyak–Viro
approach
.
7.4
. The
Kauffman
approach
.
7.5
.
Vassiliev's
invariants
coming
from the
invariant
[symbol]
.
7.6
.
Infinity
of the
number
of
long
virtual
knots
.
7.7
.
Graphs
,
chord
diagrams
and the
Kauffman
polynomial
.
7.8
.
Euler
tours
,
Gauss
circuits
and
rotating
circuits
.
7.9
. A
proof
of
Vassiliev's
conjecture
.
7.10
.
Embeddings
of
framed
4graphs

8
.
Parity
in
knot
theory
:
freeknots
:
cobordisms
.
8.1
.
Introduction
.
8.2
.
Free
knots
and
parity
.
8.3
. A
functorial
mapping
f
.
8.4
.
Invariants
.
8.5
.
Goldman's
bracket
and
Turaev's
cobracket
.
8.6
.
Applications
of
Turaev's
Delta
.
8.7
. An
analogue
of the
Kauffman
bracket
.
8.8
.
Virtual
crossing
numbers
for
virtual
knots
.
8.9
.
Cobordisms
of
free
knots

9
.
Theory
of
graphlinks
.
9.1
.
Introduction
.
9.2
.
Graphlinks
and
looped
graphs
.
9.3
.
Parity
,
minimality
and
nontrivial
examples
.
9.4
. A
generalization
of
Kauffman's
bracket
and
other
invariants
.
Minimality
theorems
.
Publisher
World Scientific Pub. Co.
Subject
Knot theory.
Identifier (Full text)
9789814401135
(electronic
bk.)
;
9789814401128
(hbk.)
;
http://www.worldscientific.com/worldscibooks/10.1142/8438#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xxv, 521 p. : ill.
Date
c2013
.
RelationIs Part Of
K & E series on knots and everything
v. 51
OCLC number
874498258
CONTENTdm number
473
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