Article

# Finite Type Invariants Of Classical And Virtual Knots

• ##### Mikhail Goussarov
Topology (Impact Factor: 0.23). 02/1970; DOI: 10.1016/S0040-9383(99)00054-3
Source: CiteSeer

ABSTRACT . We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree n in the classical sense. A universal invariant of degree n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence. 1. Virtualization Recently L. Kauffman introduced a notion of a virtual knot, extending the knot theory in an unexpected direction. We show here that this extension motiva...

0 Bookmarks
·
83 Views
• Source
##### Article: The Pure Virtual Braid Group Is Quadratic
[Hide abstract]
ABSTRACT: If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.
Selecta Mathematica 10/2011; · 0.72 Impact Factor
• Source
##### Article: Problems on invariants of knots and 3-manifolds
[Hide abstract]
ABSTRACT: This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in problem sessions in the workshop and seminars on `Invariants of Knots and 3-Manifolds' held at Kyoto in 2001.
07/2004;
• Source
##### Article: On a class of virtual knots with unit Jones polynomial
[Hide abstract]
ABSTRACT: A collection of nontrivial virtual knots with unit Jones polynomial, obtained by vir-tualizing a single crossing of a knot diagram, contains no classical knots.
Journal of Knot Theory and Its Ramifications 01/2004; · 0.40 Impact Factor