In this paper 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 which is stronger than the Homflypt polynomial. Finally, we present a closed formula for this invariant, by W.B.R. Lickorish, which uses Homflypt polynomials of sublinks and linking numbers of a given oriented link.

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In this paper we study properties of the Markov trace tr d and the specialized trace tr d,D on the Yokonuma-Hecke algebras, such as behaviour under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace tr d,D and also invariants for transverse links through the trace tr d . 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 behaviour on links.braid monoid to the Yokonuma-Hecke algebras, one can also define invariants for singular knots and links [JuLa4].In this paper we adapt the constructions in [JuLa2, JuLa3, JuLa4] for defining invariants for framed, classical and singular knots and links respectively, which are denoted by Φ d,D , Θ d,D and Ψ d,D respectively, using a different quadratic relation for the Yokonuma-Hecke algebra [ChPo]. Moreover, we construct invariants for transverse knots and links, for which no use of the Econdition is needed. Indeed, as we observe, the braid equivalence in the class of transverse knots and links requires only positive stabilization [OrSh, Wr], making the algebras Y d,n (q) a natural algebraic companion. This leads to the construction of a (d + 1)-variable transverse link invariant, M d (q, z, x 1 , . . . , x d−1 ), via the algebras Y d,n (q) and the trace tr d (see Theorem 4.14).We focus now on the class of classical knots and links. The invariants of classical knots and links defined via the Yokonuma-Hecke algebras need to be compared with other known invariants, especially with the Homflypt polynomial P (q, z). The first result in this direction was obtained in [ChLa] where it was shown that these invariants coincide with the polynomial P (q, z) only when we are in the group algebra (q = ±1) or when tr d (e i ) = 1, which is equivalent to the solution of the E-system comprising the d-th roots of unity. Further, it was shown in [ChLa] that there is no algebra homomorphism between the Yokonuma-Hecke algebra and the Iwahori-Hecke algebras which respects the trace rules, except when tr d (e i ) = 1. Moreover, it is very difficult to make a comparison of the invariants diagrammatically, since the skein relation of the framed link invariants [JuLa2, Proposition 7] has no topological meaning (see Figure 1) for the classical link invariants [JuLa3]. Yet, the classical link invariants may be topologically equivalent with the polynomial P , in the sense that they do not distinguish more or less knot and link pairs. Computations were now required, but their complexity would increase drastically due to the appearance of the e i 's in the quadratic relation. Consequently, K. Karvounis and M. Chmutov developed independently computational packages, which were cross-checked. The invariants Θ d,D were c...

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We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.