FINITE TYPE INVARIANTS FOR SINGULAR SURFACE BRAIDS ASSOCIATED WITH SIMPLE 1-HANDLE SURGERIES

2004 ◽  
Vol 13 (01) ◽  
pp. 1-11
Author(s):  
MASAHIDE IWAKIRI
Keyword(s):  
Finite Type ◽  
Singular Surface ◽  
Finite Type Invariants ◽  
Type Invariant

S. Kamada introduced finite type invariants of knotted surfaces in 4-space associated with finger moves and 1-handle surgeries. In this paper, we define finite type invariants of surface braids associated with simple 1-handle surgeries and prove that a certain set of finite type invariants controls all finite type invariants. As a consequence, we see that every finite type invariant is not a complete invariant.

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.

CLASPER-MOVES AMONG RIBBON 2-KNOTS CHARACTERIZING THEIR FINITE TYPE INVARIANTS

2006 ◽  
Vol 15 (09) ◽  
pp. 1163-1199 ◽  
Author(s):  
TADAYUKI WATANABE
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Topological Interpretation ◽  
Type Invariant

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

scholarly journals Brunnian links, claspers and Goussarov–Vassiliev finite type invariants

2007 ◽  
Vol 142 (3) ◽  
pp. 459-468 ◽  
Author(s):  
KAZUO HABIRO
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Type Invariant ◽  
Brunnian Links

AbstractGoussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

scholarly journals FINITE TYPE INVARIANTS AND MILNOR INVARIANTS FOR BRUNNIAN LINKS

2008 ◽  
Vol 19 (06) ◽  
pp. 747-766 ◽  
Author(s):  
KAZUO HABIRO ◽  
JEAN-BAPTISTE MEILHAN
Keyword(s):  
Quadratic Form ◽  
Finite Type ◽  
Finite Type Invariants ◽  
Type Invariant ◽  
Milnor Invariants ◽  
Link Homotopy ◽  
Homotopy Invariants ◽  
Brunnian Links

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, Habiro proved that the restriction to Brunnian links of any Goussarov–Vassiliev finite type invariant of (n + 1)-component links of degree < 2n is trivial. The purpose of this paper is to study the first nontrivial case. We show that the restriction of an invariant of degree 2n to (n + 1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n + 1.

scholarly journals FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants

2000 ◽  
Vol 09 (06) ◽  
pp. 735-758 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Finite Type Invariants ◽  
Homotopy Invariant ◽  
Type Invariant ◽  
Link Homotopy ◽  
Homotopy Invariants

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

scholarly journals SIMPLIFIED NUMERICAL FORM OF UNIVERSAL FINITE TYPE INVARIANT OF GAUSS WORDS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350037
Author(s):  
TOMONORI FUKUNAGA ◽  
TAKAYUKI YAMAGUCHI ◽  
TAKAAKI YAMANOI
Keyword(s):  
Normal Form ◽  
Computer Program ◽  
Finite Type ◽  
Group Structure ◽  
Complete Classification ◽  
Finite Type Invariants ◽  
Partial Classification ◽  
Type Invariant

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

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.

scholarly journals A splicing formula for the LMO invariant

2020 ◽  
pp. 1-28
Author(s):  
Gwénaël Massuyeau ◽  
Delphine Moussard
Keyword(s):  
Finite Type ◽  
Rational Homology ◽  
Three Sphere ◽  
Type Invariant ◽  
Low Degrees ◽  
Surgery Formula

Abstract We prove a “splicing formula” for the LMO invariant, which is the universal finite-type invariant of rational homology three-spheres. Specifically, if a rational homology three-sphere M is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology three-spheres, our formula expresses the LMO invariant of M in terms of the Kontsevich–LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$ . The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita’s formula for the Casson–Walker invariant, and we observe that the second term of the Ohtsuki series is not additive under “standard” splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$ , hence we get a “satellite formula” for the Kontsevich–LMO invariant.

Sign in / Sign up

Export Citation Format

Share Document