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Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants
Journal of Knot Theory and Its Ramifications ( IF 0.3 ) Pub Date : 2021-07-17
Luis Paris, Loïc Rabenda

Let Rf=[A±1] be the algebra of Laurent polynomials in the variable A and let Ra=[A±1,z1,z2,] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z1,z2,. For n1 we denote by VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(Rf)}n=1 and {ATLn(Ra)}n=1 in terms of diagrams. For each n1 we determine presentations for both, VTLn(Rf) and ATLn(Ra). We determine sequences of homomorphisms {ρnf:Rf[VBn]VTLn(Rf)}n=1 and {ρna:Ra[VBn]ATLn(Ra)}n=1, we determine Markov traces {Tnf:VTLn(Rf)Rf}n=1 and {Tna:ATLn(Ra)Ra}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 n1, the standard Temperley–Lieb algebra TLn embeds into both, VTLn(Rf) and ATLn(Ra), and that the restrictions to {TLn}n=1 of the two Markov traces coincide.



中文翻译:

虚拟和箭头 Temperley-Lieb 代数、马尔可夫迹和虚拟链接不变量

电阻F=[一种±1] 是变量中洛朗多项式的代数 一种 然后让 电阻一种=[一种±1,z1,z2,] 是变量中洛朗多项式的代数 一种 和变量中的标准多项式 z1,z2,. 为了 n1 我们表示为 VBn 虚拟编织组 n股。我们定义了两个代数塔{VTLn(电阻F)}n=1{ATLn(电阻一种)}n=1在图表方面。对于每个n1 我们为两者确定演示文稿, VTLn(电阻F)ATLn(电阻一种). 我们确定同态序列{ρnF电阻F[VBn]VTLn(电阻F)}n=1{ρn一种电阻一种[VBn]ATLn(电阻一种)}n=1,我们确定马尔可夫迹 {nFVTLn(电阻F)电阻F}n=1{n一种ATLn(电阻一种)电阻一种}n=1,并且我们表明从这些马尔可夫迹获得的虚拟链接的不变量是 F-第一道的多项式和第二道的箭头多项式。我们证明,对于每个n1, 标准 Temperley-Lieb 代数 TLn 嵌入两者, VTLn(电阻F)ATLn(电阻一种),并且限制 {TLn}n=1 两条马尔可夫迹重合。

更新日期:2021-07-19
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