VASSILIEV INVARIANTS AND DOUBLE DATING TANGLES

2002 ◽  
Vol 11 (04) ◽  
pp. 527-544 ◽  
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Finite Type ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Link Invariant ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Framed Link

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

GENERA OF KNOTS AND VASSILIEV INVARIANTS

1999 ◽  
Vol 08 (02) ◽  
pp. 253-259
Author(s):  
A. Stoimenow
Keyword(s):  
Alexander Polynomial ◽  
Maximal Degree ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Finite Set

We prove that there is no non-constant Vassiliev invariant which is constant on alternating knots of Infinitely many genera (contrasting the existence of the Conway Vassiliev invariants, which vanish on any finite set of genera) and that a (non-constant) knot invariant with values bounded by a funciton of the genus, in particular any invariant depending just on genus, signature and maximal degree of the Alexander polynomial, is not a Vassiliev invariant.

scholarly journals ON VASSILIEV INVARIANTS OF ORDER TWO FOR STRING LINKS

2005 ◽  
Vol 14 (05) ◽  
pp. 665-687 ◽  
Author(s):  
JEAN-BAPTISTE MEILHAN
Keyword(s):  
Link Invariant ◽  
Vassiliev Invariants ◽  
Linking Number ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
String Links ◽  
Combinatorial Formulas ◽  
Necessary And Sufficient

We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant V2, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is applied to the theory of claspers for string links.

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

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.

Knot Theory and Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 39-49 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Statistical Mechanics ◽  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Scattering Amplitude ◽  
Basic Structure ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Quantum Link

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

On the skein theory of dichromatic links and invariants of finite type

2017 ◽  
Vol 26 (13) ◽  
pp. 1750092 ◽  
Author(s):  
Khaled Bataineh
Keyword(s):  
Finite Type ◽  
Partial Derivative ◽  
Polynomial Invariant ◽  
Partial Derivatives ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Combinatorial Formulas ◽  
Skein Theory

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

scholarly journals THE COEFFICIENTS OF THE HOMFLYPT AND THE KAUFFMAN POLYNOMIALS ARE POINTWISE LIMITS OF VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (02) ◽  
pp. 203-215
Author(s):  
LAURE HELME-GUIZON
Keyword(s):  
Link Invariant ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Pointwise Limit

The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the Vassiliev conjecture holds in the case of the coefficients of the HOMFLY and the Kauffman polynomials.

VASSILIEV INVARIANTS AND SIMILARITY INDICES OF KNOTS, LINKS AND TANGLES

2006 ◽  
Vol 15 (09) ◽  
pp. 1201-1214 ◽  
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Open Problem ◽  
Necessary Conditions ◽  
Similarity Index ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Similarity Indices

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not. We give relations between the degrees of Vassiliev invariants and the similarity indices of knots, links and tangles. From these, we get necessary conditions for a knot invariant to be a Vassiliev invariant and get methods to detect the similarity index of two knots or tangles.

SIMILARITY INDICES OF AMPHICHEIRAL KNOTS AND TANGLES

2008 ◽  
Vol 17 (04) ◽  
pp. 483-494
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Open Problem ◽  
Necessary Conditions ◽  
Similarity Index ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Similarity Indices ◽  
Mod P

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not. Let T and S be two tangles which are n-similar for some natural number n and let the closure [Formula: see text] of T be well-defined. Let T* and S* be the mirror images of T and S respectively. Then we show that for any prime number p, [Formula: see text] mod p for any integral Vassiliev invariant v of degree ≤ np. We also show that [Formula: see text] for any Vassiliev invariant w of degree ≤ n if n is odd. Therefore, if an amphicheiral knot can be distinguished from a trivial knot by a Vassiliev invariant, then it has an even triviality index. From these, we get some necessary conditions for a knot invariant to be a Vassiliev invariant and get a method to detect the similarity index of two knots or tangles.

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