scholarly journals FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES

2010 ◽  
Vol 19 (03) ◽  
pp. 355-384 ◽  
Author(s):  
JAMES CONANT ◽  
JACOB MOSTOVOY ◽  
TED STANFORD
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Finite Type ◽  
Equivalence Classes ◽  
Connected Sum ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Conway Polynomial ◽  
Band Pass

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n 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.

KNOTS WITH GIVEN FINITE TYPE INVARIANTS AND CONWAY POLYNOMIAL

2006 ◽  
Vol 15 (02) ◽  
pp. 205-215 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Finite Type ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Conway Polynomial

It is well-known that the coefficient of zmof the Conway polynomial is a Vassiiev invariant of order m. In this paper, we show that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K.

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 Cabling the Vassiliev Invariants

1997 ◽  
Vol 06 (03) ◽  
pp. 327-358 ◽  
Author(s):  
A. Kricker ◽  
B. Spence ◽  
I. Aitchison
Keyword(s):  
Finite Type ◽  
Feynman Diagrams ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Eigenvectors And Eigenvalues

We characterise the cabling operations on the weight systems of finite type knot invariants. The eigenvectors and eigenvalues of this family of operations are described. The canonical deframing projection for these knot invariants is described over the cable eigenbasis. The action of immanent weight systems on general Feynman diagrams is considered, and the highest eigenvalue cabling eigenvectors are shown to be dual to the immanent weight systems. Using these results, we prove a recent conjecture of Bar-Natan and Garoufalidis on cablings of weight systems.

scholarly journals PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240001 ◽  
Author(s):  
MICAH WHITNEY CHRISMAN ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Finite Type ◽  
Combinatorial Formula ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Additional Information ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Gauss Diagram

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

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.

Alexander-Conway Limits of Many Vassiliev Weight Systems

1997 ◽  
Vol 06 (05) ◽  
pp. 687-714 ◽  
Author(s):  
A. Kricker
Keyword(s):  
Large Class ◽  
Vassiliev Invariants ◽  
Conway Polynomial ◽  
Simple Logic

Previous work has shown that certain leading orders of arbitrary Vassiliev invariants are generically in the algebra of the coefficients of the Alexander-Conway polynomial [KSA]. Here we illustrate this for a large class of examples, exposing the simple logic behind several existing results in the literature [FKV, BNG]. This approach facilitates an extension to a large class of Lie (super) algebras.

scholarly journals ON VASSILIEV INVARIANTS FOR LINKS IN HANDLEBODIES

1998 ◽  
Vol 07 (05) ◽  
pp. 701-712 ◽  
Author(s):  
VLADIMIR V. VERSHININ
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Special Class ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Relative Version ◽  
One To One

The notion of Vassiliev algebra in case of handlebodies is developed. Analogues of the results of J. Baez for links in handlebodies are proved. This implies that there is a one-to-one correspondence between the special class of finite type invariants of links in hanlebodies and the homogeneous Markov traces on Vassiliev algebras. This approach uses the singular braid monoid and braid group in a handlebody and the generalizations of the theorem of J. Alexander and the theorem of A. A. Markov for singular links and braids and the relative version of Markov's theorem.

Sign in / Sign up

Export Citation Format

Share Document