A framization of the Hecke algebra of type B

Abstract

Abstract. In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.

Framization of a Temperley-Lieb algebra of type B

“…Moreover, if we map the t i 's to a fixed non-trivial d-th root of the unity, we have an epimorphism from Y B d,n to H n (u, 1). In [6] are given two different linear bases for Y B d,n , denoted by D n and C n respectively. We only recall the second one, since it is the one that is used for the definition of the Markov trace over the algebra Y B d,n .…”

“…In [6] the first author together with Juyumaya and Lambropoulou proved that Y B d,n supports a unique Markov trace. In brief, this method consists in constructing a certain family of linear maps tr n : Y B d,n −→ Y B d,n−1 , called relative traces, which builds step by step the desired Markov properties (see also cf.…”

“…However, there has been a growing interest also in the framization of algebras that are related to Coxeter systems of type B. Indeed, the affine and cyclotomic Yokonuma-Hecke algebra was introduced in [1], and recently the first author together Juyumaya and Lambopoulou [6] introduced a framization of the Hecke algebra of type B, Y B d,n (u, v), a construction analogous to the Yokonuma-Hecke algebra but in the context of Coxeter systems of the type B.…”

“…In this paper we extend FTL d,n (q), the Framization of the Temperley-Lieb algebra of type A, to Coxeter groups of type B by implementing methods of [6]. To do so, we first consider the generalized Temperley-Lieb algebra that is associated to an arbitrary Coxeter system [12], and we specialize it to the case of Coxeter systems of type B.…”

“…We then proceed with the Framization of TL B n (u, v) which is defined as a quotient of the algebra Y B d,n (u, v). The main theorem determines the necessary and sufficient conditions so that the trace defined on Y B d,n (u, v) [6] passes through to the newly constructed quotient algebra. Finally, given the role of knot algebras such as the Temperley-Lieb algebra of type A and the Framization of the Temperley-Lieb algebra of type A in the construction of link invariants, it is meaningful to investigate which of the conditions of the main theorem furnish topologically non-trivial invariants for framed and classical links and we proceed to define them.…”