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Let R f = ℤ [ A ± 1 ] be the algebra of Laurent polynomials in the variable A and let R a = ℤ [ A ± 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 ≥ 1 we denote by VB n the virtual braid group on n strands. We define two towers of algebras { VTL n (R f) } n = 1 ∞ and { ATL n (R a) } n = 1 ∞ in terms of diagrams. For each n ≥ 1 we determine presentations for both, VTL n (R f) and ATL n (R a). We determine sequences of homomorphisms { ρ n f : R f [ VB n ] → VTL n (R f) } n = 1 ∞ and { ρ n a : R a [ VB n ] → ATL n (R a) } n = 1 ∞ , we determine Markov traces { T n ′ f : VTL n (R f) → R f } n = 1 ∞ and { T n ′ a : ATL n (R a) → R a } 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 ≥ 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 } n = 1 ∞ of the two Markov traces coincide. [ABSTRACT FROM AUTHOR]