THE BRAID INDEX AND THE GROWTH OF VASSILIEV INVARIANTS

1999 ◽  
Vol 08 (06) ◽  
pp. 799-813 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Upper Bounds ◽  
New Approach ◽  
Vassiliev Invariants ◽  
Braid Index ◽  
Vassiliev Invariant ◽  
Bridge Number

We use the new approach of braiding sequences to prove exponential upper bounds for the number of Vassiliev invariants on knots with bounded braid index, bounded bridge number and arborescent knots. We prove, that any Vassiliev invariant of degree k is determined by its values on knots with braid index at most k + 1.

MANY CLASSICAL KNOT INVARIANTS ARE NOT VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (01) ◽  
pp. 7-10 ◽  
Author(s):  
JOHN DEAN
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Bridge Number

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

ON FINITENESS OF VASSILIEV INVARIANTS AND A PROOF OF THE LIN-WANG CONJECTURE VIA BRAIDING POLYNOMIALS

2001 ◽  
Vol 10 (05) ◽  
pp. 769-780 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Upper Bound ◽  
Special Class ◽  
Crossing Number ◽  
New Approach ◽  
Vassiliev Invariants ◽  
A Value ◽  
Vassiliev Invariant

Using the new approach of braiding sequences we give a proof of the Lin-Wing conjecture, stating that a Vassiliev invariant ν of degree k has a value Oν (c(K)k) on a knot K, where c(K) is the crossing number of K and Oν depends on ν only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only. We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.

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.

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.

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.

A Cn-MOVE FOR A KNOT AND THE COEFFICIENTS OF THE CONWAY POLYNOMIAL

2008 ◽  
Vol 17 (07) ◽  
pp. 771-785 ◽  
Author(s):  
YOSHIYUKI OHYAMA ◽  
HARUMI YAMADA
Keyword(s):  
Vassiliev Invariants ◽  
Conway Polynomial ◽  
Vassiliev Invariant

It is shown that two knots can be transformed into each other by Cn-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a Cn-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a Cn-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

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