Finite type invariants of ribbon 2-knots

1999 ◽  
pp. 187-196 ◽  
Author(s):  
Kazuo Habiro ◽  
Taizo Kanenobu ◽  
Akiko Shima
Keyword(s):  
Finite Type ◽  
Finite Type Invariants

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′.

2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Type Invariant

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

scholarly journals Braids on surfaces and finite type invariants

2004 ◽  
Vol 338 (2) ◽  
pp. 157-162 ◽  
Author(s):  
Paolo Bellingeri ◽  
Louis Funar
Keyword(s):  
Finite Type ◽  
Finite Type Invariants

scholarly journals Finite-type invariants of classical and virtual knots

Topology ◽  
2000 ◽  
Vol 39 (5) ◽  
pp. 1045-1068 ◽  
Author(s):  
Mikhail Goussarov ◽  
Michael Polyak ◽  
Oleg Viro
Keyword(s):  
Finite Type ◽  
Virtual Knots ◽  
Finite Type Invariants
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