虚箭头Temperley-Lieb代数、Markov迹和虚链不变量
Virtual an arrow Temperley--Lieb algebras, Markov traces, and virtual link invariants
摘要：
设Rf = Z[A $\pm$ 1 ]是关于变量A的Laurent多项式的代数，设Ra = Z[A $\pm$ 1，z1，z2，. .. ]是变量A中的洛朗多项式和变量z 1，z 2，. .. .对于n $\ge$ 1，我们用VB n表示n股上的虚拟辫群。用图的方法定义了两个代数塔{ATLn（Rf）} $\infty$ n=1和{ATLn（Ra）} $\infty$ n=1.对于每个n $\ge$ 1，我们确定了两个的表示，即RNN（R f）和ATL n（R a）。我们确定了同态序列{ $ρ$ f n：Rf [VB n ] $\rightarrow$ n（Rf）} $\infty$ n=1和{ $ρ$ a n：Ra [VB n ] $\rightarrow$ ATL n（Ra）} $\infty$ n=1，我们确定了Markov迹{T f n：n（Rf）$\rightarrow$ R f } $\infty$ n=1和{T a n：ATL n（R a）$\rightarrow$ R a } $\infty$ n=1，证明了由这些Markov迹得到的虚链路的不变量对于第一条迹是f多项式，对于第二条迹是箭头多项式.证明了对于每个n个$\ge$ 1，标准Temperley-Lieb代数TL n嵌入到两个Markov迹中，并且对{TL n } $\infty$ n=1的限制是一致的.
英文摘要：
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 {$ρ$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\infty$ n=1 and {$ρ$ 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.
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