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Let R f = Z[A $\pm$1 ] be the algebra of Laurent polynomials in the variableA and let R a = Z[A $\pm$1 , z 1 , z 2 ,. .. ] be the algebra of Laurentpolynomials in the variable A and standard polynomials in the variables z 1 , z2 ,. .. . 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 determinepresentations for both, VTL n (R f) and ATL n (R a). We determine sequences ofhomomorphisms {$\rho$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\infty$ n=1and {$\rho$ a n : R a [VB n ] $\rightarrow$ ATL n (R a)} $\infty$ n=1 , wedetermine Markov traces {T f n : VTL n (R f) $\rightarrow$ R f } $\infty$ n=1and {T a n : ATL n (R a) $\rightarrow$ R a } $\infty$ n=1 , and we show thatthe invariants for virtual links obtained from these Markov traces are thef-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 nembeds into both, VTL n (R f) and ATL n (R a), and that the restrictions to {TLn } $\infty$ n=1 of the two Markov traces coincide.

Virtual an arrow Temperley--Lieb algebras, Markov traces, and virtual link invariants