In this paper we first discuss a Temperley-Lieb algebra associated to the Coxeter group of type $\mathtt{B}$ which is the natural extension of the classical case, in the sense that it can be expressed as a quotient of the Hecke algebra of type B over an appropriate two-sided ideal. We then give the necessary and sufficient conditions so that the Markov trace defined on the Hecke algebra of type $\mathtt{B}$ factors through to the quotient algebra and we construct the corresponding knot invariants. Next, following the results recently obtained for groups of type $\mathtt{A}$, we define a framization of such a Temperley-Lieb algebra as a proper quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass through to the framization quotient algebra. Finally, we present the derived invariants for framed and classical knots and links inside th 查看全文>>