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

TL;DR

This paper introduces algebraic structures called virtual Temperley--Lieb algebras and arrow Temperley--Lieb algebras, constructs Markov traces on them, and shows how they produce invariants for virtual links, including the f-polynomial and arrow polynomial.

Contribution

It defines new algebraic towers related to virtual braids, provides presentations, and establishes Markov traces that yield virtual link invariants, connecting to known polynomials.

Findings

Constructed two towers of algebras for virtual braids.

Defined Markov traces leading to virtual link invariants.

Established embeddings of Temperley-Lieb algebra into these structures.

Abstract

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.