Classical link invariants from the framizations of the Iwahori-Hecke algebra and the Temperley-Lieb algebra of type A

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras Yd,n(q), which are not topologically equivalent to the Homflypt polynomial. We then present the algebra FTLd,n(q) which is the appropriate Temperley-Lieb analogue of Yd,n(q), as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma-Hecke algebra, and also its quotient, the partition Temperley-Lieb algebra PTLn(q) and we prove an isomorphism of this algebra with a subalgebra of FTLd,n(q).