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Article Dans Une Revue Journal of Knot Theory and Its Ramifications Année : 2021

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.

Résumé

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.
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Dates et versions

hal-03467039 , version 1 (06-12-2021)

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Luis Paris, Loïc Rabenda. Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants. Journal of Knot Theory and Its Ramifications, 2021, 30 (06), pp.2150041. ⟨10.1142/S0218216521500413⟩. ⟨hal-03467039⟩
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