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.

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.

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.

Hidden structures of knot invariants

2014 ◽  
Vol 29 (29) ◽  
pp. 1430063 ◽  
Author(s):  
Alexey Sleptsov
Keyword(s):  
Symmetric Group ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Double Affine Hecke Algebra ◽  
Loop Expansion ◽  
Group Characters ◽  
The Family ◽  
Homfly Polynomials ◽  
Genus Expansion

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

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.

Weights of Feynman diagrams, link polynomials and Vassiliev knot invariants

1995 ◽  
Vol 04 (01) ◽  
pp. 163-188 ◽  
Author(s):  
Sergey Piunikhin
Keyword(s):  
Perturbation Theory ◽  
Irreducible Representation ◽  
Hopf Algebra ◽  
Lie Algebras ◽  
Feynman Diagrams ◽  
Primitive Elements ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Chern Simons

We prove that the construction of Vassiliev invariants by expanding the link polynomials Pg,V(q, q−1) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified. We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones. The explicit description of primitive elements in the Kontsevich Hopf algebra is given.

scholarly journals Bounds on übercrossing and petal numbers for knots

2015 ◽  
Vol 24 (02) ◽  
pp. 1550012 ◽  
Author(s):  
Colin Adams ◽  
Orsola Capovilla-Searle ◽  
Jesse Freeman ◽  
Daniel Irvine ◽  
Samantha Petti ◽  
...  
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Bridge Number

An n-crossing is a point in the projection of a knot where n strands cross so that each strand bisects the crossing. An übercrossing projection has a single n-crossing and a petal projection has a single n-crossing such that there are no loops nested within others. The übercrossing number, ü(K), is the smallest n for which we can represent a knot K with a single n-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the übercrossing number and petal number to well known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are sharp for (r, r + 1)-torus knots. We also explore the behavior of übercrossing number under composition.

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.

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.

scholarly journals RIBBONLENGTH OF TORUS KNOTS

2008 ◽  
Vol 17 (01) ◽  
pp. 13-23 ◽  
Author(s):  
BROOKE KENNEDY ◽  
THOMAS W. MATTMAN ◽  
ROBERTO RAYA ◽  
DAN TATING
Keyword(s):  
Crossing Number ◽  
Regular Polygons ◽  
Torus Knots ◽  
Torus Knot

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1) cot (π/(2q + 1)) (respectively, 2q cot (π/(2q + 1))). Using these calculations, we provide the bounds c1 ≤ 2/π and c2 ≥ 5/3 cot π/5 for the constants c1 and c2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c1 C(K) ≤ R(K) ≤ c2 C(K).

Linearized chord diagrams and an upper bound for vassiliev invariants

2000 ◽  
Vol 09 (07) ◽  
pp. 847-853 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Upper Bound ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Chord Diagrams

Recently, Stoimenow [J. Knot Th. Ram. 7 (1998), 93–114] gave an upper bound on the dimension dn of the space of order n Vassiliev knot invariants, by considering chord diagrams of a certain type. We present a simpler argument which gives a better bound on the number of these chord diagrams, and hence on dn.

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