scholarly journals Homfly polynomials of generalized Hopf links

2002 ◽  
Vol 2 (1) ◽  
pp. 11-32 ◽  
Author(s):  
Hugh R Morton ◽  
Richard J Hadji
Keyword(s):  
Homfly Polynomials ◽  
Hopf Links

HOMFLY Polynomials of Some Generalized Hopf Links

2000 ◽  
Vol 09 (07) ◽  
pp. 865-883 ◽  
Author(s):  
TAT-HUNG CHAN
Keyword(s):  
Explicit Formulas ◽  
Recurrence Equations ◽  
Homfly Polynomials ◽  
Hopf Links ◽  
Hopf Link

The Hopf link, consisting of two unknots wrapped around each other, is the simplest possible nontrivial link with more than one component. We can generalize it to two bundles of "parallel" unknots wrapped around each other. In this paper, we show that when one of the two bundles has a fixed side, the HOMFLY polynomials of the links satisfy a system of recurrence equations. This leads to a procedure for computing explicit formulas for the HOMFLY polynomials.

scholarly journals Khovanov homology detects the Hopf links

2019 ◽  
Vol 26 (5) ◽  
pp. 1281-1290
Author(s):  
John A. Baldwin ◽  
Steven Sivek ◽  
Yi Xie
Keyword(s):  
Khovanov Homology ◽  
Hopf Links

scholarly journals On the block structure of the quantum ℛ-matrix in the three-strand braids

2018 ◽  
Vol 33 (17) ◽  
pp. 1850105 ◽  
Author(s):  
L. Bishler ◽  
An. Morozov ◽  
Sh. Shakirov ◽  
A. Sleptsov
Keyword(s):  
Block Structure ◽  
Representation Formula ◽  
Building Blocks ◽  
Essential Condition ◽  
The Road ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
On The Road ◽  
Homfly Polynomials ◽  
Block Diagonal

Quantum [Formula: see text]-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation [Formula: see text] of [Formula: see text] associated with each strand, one needs two matrices: [Formula: see text] and [Formula: see text]. They are related by the Racah matrices [Formula: see text]. Since we can always choose the basis so that [Formula: see text] is diagonal, the problem is reduced to evaluation of [Formula: see text]-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that [Formula: see text]-matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of [Formula: see text] matrix. In this case in order to get a block-diagonal matrix, one should rotate the [Formula: see text] defined by the Racah matrix in the accidental sector by the angle exactly [Formula: see text].

scholarly journals On colored HOMFLY polynomials for twist knots

2014 ◽  
Vol 29 (34) ◽  
pp. 1450183 ◽  
Author(s):  
Andrei Mironov ◽  
Alexei Morozov ◽  
Andrey Morozov
Keyword(s):  
Matrix Model ◽  
Initial Conditions ◽  
The Family ◽  
Differential Expansion ◽  
Generic Representation ◽  
Homfly Polynomials ◽  
Model Approach

Recent results of Gu and Jockers provide the lacking initial conditions for the evolution method in the case of the first nontrivially colored HOMFLY polynomials H[21] for the family of twist knots. We describe this application of the evolution method, which finally allows one to penetrate through the wall between (anti)symmetric and non-rectangular representations for a whole family. We reveal the necessary deformation of the differential expansion, what, together with the recently suggested matrix model approach gives new opportunities to guess what it could be for a generic representation, at least for the family of twist knots.

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.

scholarly journals Cabling procedure for the colored HOMFLY polynomials

2014 ◽  
Vol 178 (1) ◽  
pp. 1-58 ◽  
Author(s):  
A. S. Anokhina ◽  
A. A. Morozov
Keyword(s):  
Homfly Polynomials

scholarly journals A relation between the Kauffman and the HOMFLY polynomials for torus knots

1996 ◽  
Vol 37 (4) ◽  
pp. 2013-2042 ◽  
Author(s):  
J. M. F. Labastida ◽  
E. Pérez
Keyword(s):  
Torus Knots ◽  
Homfly Polynomials
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