  Let R f = Z[A $\pm$1 ] be the algebra of Laurent polynomials in the variable
A and let R a = Z[A $\pm$1 , z 1 , z 2 ,. .. ] be the algebra of Laurent
polynomials in the variable A and standard polynomials in the variables z 1 , z
2 ,. .. . For n $\ge$ 1 we denote by VB n the virtual braid group on n strands.
We define two towers of algebras {VTL n (R f)} $\infty$ n=1 and {ATL n (R a)}
$\infty$ n=1 in terms of diagrams. For each n $\ge$ 1 we determine
presentations for both, VTL n (R f) and ATL n (R a). We determine sequences of
homomorphisms {$\rho$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\infty$ n=1
and {$\rho$ a n : R a [VB n ] $\rightarrow$ ATL n (R a)} $\infty$ n=1 , we
determine Markov traces {T f n : VTL n (R f) $\rightarrow$ R f } $\infty$ n=1
and {T a n : ATL n (R a) $\rightarrow$ R a } $\infty$ n=1 , and we show that
the invariants for virtual links obtained from these Markov traces are the
f-polynomial for the first trace and the arrow polynomial for the second trace.
We show that, for each n $\ge$ 1, the standard Temperley-Lieb algebra TL n
embeds into both, VTL n (R f) and ATL n (R a), and that the restrictions to {TL
n } $\infty$ n=1 of the two Markov traces coincide.
