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.

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.

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.

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.

SATANIC AND THELEMIC CIRCLES ON KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350017 ◽  
Author(s):  
G. FLOWERS
Keyword(s):  
Geometric Interpretation ◽  
Second Order ◽  
Geometric Properties ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials

2007 ◽  
Vol 59 (2) ◽  
pp. 418-448 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Open Problem ◽  
Negative Answer ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Homfly Polynomials

AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomial Q contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from Q is an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree d ≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

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.

scholarly journals An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

2003 ◽  
Vol 12 (06) ◽  
pp. 767-779 ◽  
Author(s):  
Jörg Sawollek
Keyword(s):  
Virtual Link ◽  
Zeroth Order ◽  
Opposite Orientation ◽  
Vassiliev Invariants ◽  
Virtual Knot ◽  
First Order ◽  
Virtual Knots ◽  
Conway Polynomial ◽  
Vassiliev Invariant ◽  
Open Question

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

LOCAL MOVES AND GORDIAN COMPLEXES

2006 ◽  
Vol 15 (09) ◽  
pp. 1215-1224 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Vassiliev Invariants ◽  
Conway Polynomial ◽  
Vassiliev Invariant ◽  
The Relationship

By the works of Gusarov [2] and Habiro [3], it is known that a local move called the Cnmove is strongly related to Vassiliev invariants of order less than n. The coefficient of the znterm in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and KCnthe set of knots obtained from a knot K by a single Cnmove. Let [Formula: see text] be the set of the Conway polynomials [Formula: see text] for a set of knots [Formula: see text]. Our main result is the following: There exists a pair of knots K1, K2such that ∇K1= ∇K2and [Formula: see text]. In other words, the CnGordian complex is not homogeneous with respect to Conway polynomial.

REPRESENTATIONS OF KNOT GROUPS AND VASSILIEV INVARIANTS

1996 ◽  
Vol 05 (04) ◽  
pp. 421-425 ◽  
Author(s):  
DANIEL ALTSCHULER
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Torus Knots ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
A Finite Group

We show that the number of homomorphisms from a knot group to a finite group G cannot be a Vassiliev invariant, unless it is constant on the set of (2, 2p+1) torus knots. In several cases, such as when G is a dihedral or symmetric group, this implies that the number of homomorphisms is not a Vassiliev invariant.

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