Virtual and 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 VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(R-f)}(n=1)(infinity) and {ATL(n)(R-a)}(n=1)(infinity) in terms of diagrams. For each n >= 1 we determine presentations for both, VTLn(R-f) and ATL(n)(R-a). We determine sequences of homomorphisms {rho(f)(n):R-f[VBn] -> VTLn(R-f)}(n=1)(infinity) and {rho(a)(n):R-a[VBn] -> ATL(n)(R-a)}(n=1)(infinity), we determine Markov traces {T-n'(f):VTLn(R-f) -> R-f}(n=1)(infinity) and {T-n'(a):ATL(n)(R-a) -> R-a}(n=1)(infinity), 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 TLn embeds into both, VTLn(R-f) and ATL(n)(R-a), and that the restrictions to {TLn}(n=1)(infinity) of the two Markov traces coincide.