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.

The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to [6], we explore further the structures of the p-adic framed braids and the p-adic Yokonuma-Hecke algebras constructed in [6], by means of dense sub-structures approximating p-adic elements. We also construct a p-adic Markov trace on the p-adic Yokonuma-Hecke algebras and we approximate the values of the p-adic trace on p-adic elements. Surprisingly, the Markov traces do not re-scale directly to yield isotopy invariants of framed links. This leads to imposing the 'E-condition' on the trace parameters. For solutions of the 'E-system' we then define 2-variable isotopy invariants of modular framed links. These lift to p-adic isotopy invariants of classical framed links. The Yokonuma-Hecke algebras have topological interpretations in the context of framed knots, of classical knots of singular knots and of transverse knots.1991 Mathematics Subject Classification. 57M27, 20F38, 20F36, 20C08.

It is proved that the Yokonuma-Hecke algebra supports a Markov trace.

In this paper we introduce the tied links, i.e. ordinary links provided with some 'ties' between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpreted as the usual generators of the braid algebra, and the other half can be interpreted as ties between consecutive strands; this interpretation leads to the definition of tied braids. We define an invariant polynomial for the tied links via a skein relation. Furthermore, we introduce the monoid of tied braids and we prove the corresponding theorems of Alexander and Markov for tied links. Finally, we prove that the invariant of tied links we defined can be obtained also by using the Jones recipe.

We prove that the so-called algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.1991 Mathematics Subject Classification. 57M25, 20C08, 20F36.

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In this paper we represent the classical braids in the Yokonuma-Hecke and the adelic Yokonuma-Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma-Hecke algebras, in analogy to the p-adic framed braids and the p-adic Yokonuma-Hecke algebras introduced in [4,5]. We further construct an adelic Markov trace, analogous to the p-adic Markov trace constructed in [5], and using the traces in [3] and the adelic Markov trace we define topological invariants of classical knots and links, upon imposing some condition. Each invariant satisfies a cubic skein relation coming from the Yokonuma-Hecke algebra.1991 Mathematics Subject Classification. 57M27, 20F38, 20F36, 20C08.

Abstract. 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 SB n 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).