Oriented Quantum Algebras and Invariants of Knots and Links (original) (raw)

Quantum Algebra Structures onn×nMatrices

Journal of Algebra, 1999

Quantum algebras are generalizations of quasitriangular Hopf algebras and as such are used to construct invariants of 1᎐1 tangles and in some cases of knots and 3-manifolds. In this paper we develop a general description of quantum algebra structures whose underlying algebra is the Ž . algebra M k of all n = n matrices over a field k. An important ingredin ent in the structure of a quantum algebra in this case is a solution Ž . 2 R g M k to the quantum Yang᎐Baxter equation. n We use our general results and the classification of solutions to the Ž . Ž . Ž . quantum Yang᎐Baxter equation R g M k s M k m M k found in 4 2 2

The Quantum sl<sub>2</sub>-Invariant of a Family of Knots

Applied Mathematics, 2014

We give a general formula of the quantum 2 sl-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra 2 g sl = in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum invariant.

Handlebody-knot invariants derived from unimodular Hopf algebras

Journal of Knot Theory and Its Ramifications, 2014

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.

On Knots, Complements, and 6j-Symbols

Annales Henri Poincaré, 2021

This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, SO(N) quantum 6j-symbols and (a, t)-deformed F K. First, we present a simple rule of grading change which allows us to obtain the [r]-colored quadruply-graded Kauffman homology from the [r 2 ]-colored quadruply-graded HOMFLY-PT homology for thin knots. This rule stems from the isomorphism of the representations (so 6 , [r]) ∼ = (sl 4 , [r 2 ]). Also, we find the relationship among A-polynomials of SO and SU-type coming from a differential on Kauffman homology. Second, we put forward a closed-form expression of SO(N)(N ≥ 4) quantum 6j-symbols for symmetric representations, and calculate the corresponding SO(N) fusion matrices for the cases when representations R = ,. Third, we conjecture closed-form expressions of (a, t)-deformed F K for the complements of double twist knots with positive braids. Using the conjectural expressions, we derive t-deformed ADO polynomials.

Quantum Invariants of Knotoids

Communications in Mathematical Physics

In this paper, we construct quantum invariants for knotoid diagrams in R 2. The diagrams are arranged with respect to a given direction in the plane (Morse knotoids). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.

The Homfly invariant of closed tangles

In the papers [3, 4], the author constructed a complete set of irreducible representations of the Hecke category. These representations also define invariants of oriented tangles. Since links are special examples of the oriented tangles, their invariant, which eventually coincides with the HOMFLY invariant of links, can be calculated by the same method. In this article, we calculate the invariant of the Hopf link, the Whitehead link and the Borromean link by this new method.