scholarly journals Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

2019 ◽  
Vol 30 (05) ◽  
pp. 1950021
Author(s):  
Benjamin Audoux ◽  
Delphine Moussard
Keyword(s):  
Finite Type ◽  
Rational Homology ◽  
Direct Sums ◽  
Finite Type Invariants ◽  
Null Lagrangian

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of [Formula: see text]-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

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.

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.

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