Tied links in the solid torus

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in [Formula: see text] previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type [Formula: see text] and the Markov trace defined on this.

Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

On the link invariants from the Yokonuma–Hecke algebras

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

Markov trace on a tower of affine Temperley–Lieb algebras of type Ã

We define a tower of affine Temperley–Lieb algebras of type Ã. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov–Ocneanu–Jones trace on the tower of Temperley–Lieb algebras of type A. We define an invariant of special kind of links as an application of this trace.

A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

G-LINKS INVARIANTS, MARKOV TRACES AND THE SEMI-CYCLIC Uq𝔰𝔩(2)-MODULES

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin–Turaev link invariant construction to tangles with a flat connection in a principal G-bundle of the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to 𝔰𝔩(2), defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

A STATE-SUM FORMULA FOR THE ALEXANDER POLYNOMIAL

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori–Hecke algebras of type A.

AN INVARIANT FOR SINGULAR KNOTS

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.