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

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 f n : R f [VB n ] → VTL n (R f)} ∞ n=1 and {ρ a n : R a [VB n ] → ATL n (R a)} ∞ n=1 , we determine Markov traces {T f n : VTL n (R f) → R f } ∞ n=1 and {T a n : 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