SOME REMARKS ON TABACHNIKOV’S INVARIANTS OF PLANE CURVES

1996 ◽  
Vol 05 (06) ◽  
pp. 849-857
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Plane Curves ◽  
Polynomial Invariant ◽  
Finite Type Invariants

We construct a generalization of the “free group valued” Tabachnikov’s invariant of long curves. We characterize Tabachnikov’s polynomial invariant of plane curves via a representation of the twin group. We also show that these invariants are combinations of finite type invariants.

scholarly journals Free groups and finite-type invariants of pure braids

2002 ◽  
Vol 132 (1) ◽  
pp. 117-130 ◽  
Author(s):  
JACOB MOSTOVOY ◽  
SIMON WILLERTON
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Free Groups ◽  
Braid Groups ◽  
Point Of View ◽  
Magnus Expansion ◽  
Vassiliev Invariants ◽  
Integer Coefficients ◽  
Finite Type Invariants ◽  
Theoretic Point

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

scholarly journals ARRANGEMENTS OF NESTED CURVES

2007 ◽  
Vol 16 (02) ◽  
pp. 217-225
Author(s):  
EMMANUEL FERRAND
Keyword(s):  
Finite Type ◽  
Grothendieck Group ◽  
Equivalence Classes ◽  
Plane Curves ◽  
Semi Group ◽  
Finite Type Invariants ◽  
Two Factors ◽  
Theory Of Invariants

Motivated by Arnold's theory of invariants of plane curves, we introduce the semi-group of equivalence classes of arrangements of nested curves. There exists a natural invariant of plane curves without inverse self-tangencies with values in this semi-group. We show that the associated Grothendieck group is ℤ × ℤ. These two factors correspond to previously known invariants of plane curves without inverse self-tangencies, namely Whitney's index and Arnold's J- invariant. We show that arrangements of nested curves are not classified by their finite type invariants.

scholarly journals θ-CURVE POLYNOMIALS AND FINITE-TYPE INVARIANTS

2002 ◽  
Vol 11 (04) ◽  
pp. 555-564 ◽  
Author(s):  
YOUNGSIK HUH ◽  
GYO TAEK JIN
Keyword(s):  
Finite Type ◽  
Type A ◽  
Polynomial Invariant ◽  
Finite Type Invariants ◽  
Polynomial Formula ◽  
Original Formula

The normalized Yamada polynomial, [Formula: see text], is a polynomial invariant in variable A for θ-curves. In this work, we show that the coefficients of [Formula: see text] which is obtained by replacing A with ex = ∑ xn/n! are finite-type invariants for θ-curves although the coefficients of original [Formula: see text] are not finite-type. A similar result can be obtained in the case of Yokota polynomial for θ-curves.

ON FINITE TYPE 3-MANIFOLD INVARIANTS I

1996 ◽  
Vol 05 (04) ◽  
pp. 441-461 ◽  
Author(s):  
STAVROS GAROUFALIDIS
Keyword(s):  
Vector Space ◽  
Finite Type ◽  
Commutative Algebra ◽  
Integral Homology ◽  
Knot Invariants ◽  
Finite Type Invariants ◽  
Definition Of ◽  
Type 3

Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (03) ◽  
pp. 391-405 ◽  
Author(s):  
ROLLAND TRAPP
Keyword(s):  
Finite Type ◽  
Polynomial Growth ◽  
Difference Sequence ◽  
Uniform Limit ◽  
Topological Information ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Vassiliev Invariant

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

On finite type 3-manifold invariants IV: comparison of definitions

1997 ◽  
Vol 122 (2) ◽  
pp. 291-300 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
JEROME LEVINE
Keyword(s):  
Finite Type ◽  
Integral Homology ◽  
Finite Type Invariants ◽  
Type 3

The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key lemma we compare two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S3, thus answering question 2 from [Ga].

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MYEONG-JU JEONG
Keyword(s):  
Lower Bound ◽  
Finite Type ◽  
Finite Sequence ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Combinatorial Representations ◽  
Congruent Modulo

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

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