2020/09/17 by Luis Paris, Paris, Luis, Loïc Rabenda +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2009.08170
openalex publication_date 2020/09/17 · openalex created_date 2020/09/21 · openalex updated_date 2026/07/28
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.