Modular framization of the BMW algebra

Abstract: In this work we introduce the concept of Modular Framization or simply Framization. We construct a framization $F_{d,n}$ of the Birman--Wenzl--Murakami algebra, also known as BMW algebra, and start a systematic study of this framization. We show that $F_{d,n}$ is finite dimensional and the \lq braid generators\rq\ of this algebra satisfy a quartic relation which is of minimal degree not containing the generators $t_i$. They also satisfy a quintic relation, as the smallest closed relation. We conjecture that the algebras $F_{d,n}$ support a Markov trace which allow to define polynomial invariants for unoriented knots in an analogous way that the Kauffman polynomial is derived from the BMW algebra. The idea originates from the Yokonuma--Hecke algebra, built from the classical Hecke algebra by adding framing generators and changing the Hecke algebra quadratic relation by a new quadratic relation which involves the framing generators. Using the Yokonuma--Hecke algebras and a Markov trace constructed on them\cite{ju} we produced invariants for oriented framed knots\cite{jula1,jula3}, classical knots\cite{jula4} (which satisfies a cubic skein relation), and singular knots\cite{jula2}. Moreover, we have connections with virtual braids and transversal knot theory. In this note we also apply the framization mechanism on other knot algebras, such as the Temperley--Lieb algebra, the singular Hecke algebra introduced by Paris and Rabenda and Hecke algebras of type $B$.