The HOMFLY polynomials of odd polyhedral links

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].

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The coefficients of the HOMFLY polynomials of links

Topology and its Applications ◽

10.1016/j.topol.2019.106881 ◽

2019 ◽

Vol 267 ◽

pp. 106881

Author(s):

Zhi-Xiong Tao

Keyword(s):

Homfly Polynomials

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On colored HOMFLY polynomials for twist knots

Modern Physics Letters A ◽

10.1142/s0217732314501831 ◽

2014 ◽

Vol 29 (34) ◽

pp. 1450183 ◽

Cited By ~ 20

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.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2007-018-0 ◽

2007 ◽

Vol 59 (2) ◽

pp. 418-448 ◽

Cited By ~ 8

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.

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