scholarly journals A Novel Symmetry of Colored HOMFLY Polynomials Coming from $$\mathfrak {sl}(N|M)$$ Superalgebras

2021 ◽  
Author(s):  
V. Mishnyakov ◽  
A. Sleptsov ◽  
N. Tselousov
Keyword(s):  
Homfly Polynomials

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

scholarly journals HOMFLY polynomials in representation [3, 1] for 3-strand braids

2016 ◽  
Vol 2016 (9) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
An. Morozov ◽  
A. Sleptsov
Keyword(s):  
Homfly Polynomials

scholarly journals CHARACTER EXPANSION FOR HOMFLY POLYNOMIALS III: ALL 3-STRAND BRAIDS IN THE FIRST SYMMETRIC REPRESENTATION

2012 ◽  
Vol 27 (19) ◽  
pp. 1250099 ◽  
Author(s):  
H. ITOYAMA ◽  
A. MIRONOV ◽  
A. MOROZOV ◽  
AND. MOROZOV
Keyword(s):  
General Formula ◽  
High Energy Phys ◽  
High Energy ◽  
Reflection Symmetry ◽  
Symmetric Representation ◽  
Knot Invariants ◽  
Time Variables ◽  
Close Relatives ◽  
Figure Eight Knot ◽  
Homfly Polynomials

We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a1, b1, a2, b2, …) and for the first nonfundamental representation R = [2]. Instead of calculating the mixing matrices directly, as suggested [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654], we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1, 1]. The result applies, in particular, to the figure eight knot 41, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in H. Itoyama, A. Mironov, A. Morozov and And. Morozov, arXiv:1203.5978.

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