CASSON KNOT INVARIANTS OF PERIODIC KNOTS WITH RATIONAL QUOTIENTS

2007 ◽  
Vol 16 (04) ◽  
pp. 439-460 ◽  
Author(s):  
HEE JEONG JANG ◽  
SANG YOUL LEE ◽  
MYOUNGSOO SEO
Keyword(s):  
Normal Form ◽  
Recurrence Formula ◽  
Alexander Polynomial ◽  
Knot Invariants ◽  
Knot Invariant

We give a formula for the Casson knot invariant of a p-periodic knot in S3 whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n1, -2, 2n2, …, 2n2m, 2) via the integers p, n1, n2, …, n2m(p ≥ 2 and m ≥ 1). As an application, for any integers n1, n2, ≥, n2m with the same sign, we determine the Δ-unknotting number of a p-periodic knot in S3 whose quotient is a 2-bridge link C(2, 2n1, -2, 2n2, ≥, 2n2m, 2) in terms of p, n1, n2, ≥, n2m. In addition, a recurrence formula for calculating the Alexander polynomial of the 2-bridge knot with Conway's normal form C(2n1, 2n2, ≥, 2nm) via the integers n1,n2, ≥, nm is included.

CASSON KNOT INVARIANTS OF PERIODIC KNOTS WITH RATIONAL QUOTIENTS II

2008 ◽  
Vol 17 (08) ◽  
pp. 905-923 ◽  
Author(s):  
SANG YOUL LEE ◽  
MYOUNGSOO SEO
Keyword(s):  
Normal Form ◽  
Knot Invariants ◽  
Knot Invariant

For any integers n1, n2, …, nm, we give a formula for the Casson knot invariant of a p-periodic knot (p ≥ 2) whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n1 + 1, -2, 2n2 + 1, …, 2nm + 1, (-1)m2).

scholarly journals Regional knot invariants

2017 ◽  
Vol 26 (06) ◽  
pp. 1742006
Author(s):  
Zhiqing Yang
Keyword(s):  
Alexander Polynomial ◽  
Link Diagram ◽  
Polynomial Invariant ◽  
Knot Invariants ◽  
Planar Region ◽  
Knot Invariant ◽  
The Matrix

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is called a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derived from the matrix just like the Alexander polynomial case.

scholarly journals Kontsevich’s integral for the Kauffman polynomial

1996 ◽  
Vol 142 ◽  
pp. 39-65 ◽  
Author(s):  
Thang Tu Quoc Le ◽  
Jun Murakami
Keyword(s):  
Finite Type ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

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.

Sharp Bounds on Some Classical Knot Invariants

2003 ◽  
Vol 12 (06) ◽  
pp. 805-817
Author(s):  
C. Kearton ◽  
S. M. J. Wilson
Keyword(s):  
Alexander Polynomial ◽  
Sharp Bounds ◽  
Knot Invariants ◽  
Bridge Number

There are obvious inequalities relating the Nakanishi index of a knot, the bridge number, the degree 2n of the Alexander polynomial and the length of the chain of Alexander ideals. We give examples for every positive value of n to show that these bounds are sharp.

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 Explicit formulae for two-bridge knot polynomials

2005 ◽  
Vol 78 (2) ◽  
pp. 149-166 ◽  
Author(s):  
Shinji Fukuhara
Keyword(s):  
Normal Form ◽  
Open Problem ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Elementary Number ◽  
Conway Polynomial ◽  
Coprime Integers ◽  
Explicit Formulae ◽  
Knots And Links

AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

The application of numerical topological invariants in simulations of knotted rings: A comprehensive Monte Carlo approach

2020 ◽  
pp. 2150005
Author(s):  
Franco Ferrari ◽  
Yani Zhao
Keyword(s):  
Monte Carlo ◽  
Discrete Version ◽  
Topological Invariants ◽  
One Dimensional ◽  
Knot Invariants ◽  
Knot Invariant ◽  
Sampling Process ◽  
Energy Domain ◽  
Sharp Corners ◽  
Monte Carlo Framework

In this work, a general Monte Carlo framework is proposed for applying numerical knot invariants in simulations of systems containing knotted one-dimensional ring-shaped objects like polymers and vortex lines in fluids, superfluids or other quantum liquids. A general prescription for smoothing the sharp corners appearing in discrete knots consisting of segments joined together is provided. Smoothing is very important for the correct evaluation of numerical knot invariants. A discrete version of framing is adopted in order to eliminate singularities that are possibly arising when computing the invariants. The presented algorithms for smoothing, eliminating potentially dangerous singularities and speeding up the calculations are quite general and can be applied to any discrete knot defined off- or on-lattice. This is one of the first attempts to use numerical knot invariants in order to avoid potential topology breakings during the sampling process taking place in computer simulations, in which millions of knot conformations are randomly generated. As an application, the energy domain of knotted polymer rings subjected to short-range interactions is studied using the so-called Vassiliev knot invariant of degree 2.

scholarly journals Knot invariants arising from homological operations on Khovanov homology

2016 ◽  
Vol 25 (03) ◽  
pp. 1640012 ◽  
Author(s):  
Krzysztof K. Putyra ◽  
Alexander N. Shumakovitch
Keyword(s):  
Homology Theory ◽  
Khovanov Homology ◽  
Unified Theory ◽  
Present Evidence ◽  
Knot Invariants ◽  
Knot Invariant

We construct an algebra of nontrivial homological operations on Khovanov homology with coefficients in [Formula: see text] generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the even and odd Khovanov homology combined. The case of reduced Khovanov homology is also considered.

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 413-422 ◽  
Author(s):  
WAYNE H. STEVENS
Keyword(s):  
Recursion Formula ◽  
Alexander Polynomial ◽  
Square Roots ◽  
Knot Invariants ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Recursion Formulas ◽  
Branched Cyclic Covers ◽  
Linear Recursion ◽  
Fold Cover

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

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