Let $R^f=\mathbb{Z}[A^{\pm 1}]$ be the algebra of Laurent polynomials in the variable $A$ and let $R^a=\mathbb{Z}[A^{\pm 1},z_1,z_2,\dots ]$ be the algebra of Laurent polynomials in the variable $A$ and standard polynomials in the variables $z_1,z_2,\dots .$ For $n\ge 1$ we denote by ${\mathrm{\text{VB}}}_n$ the virtual braid group on $n$ strands. We define two towers of algebras ${\{{\mathrm{\text{VTL}}}_n(R^f)\}}_{n=1}^{\infty }$ and ${\{{\mathrm{\text{ATL}}}_n(R^a)\}}_{n=1}^{\infty }$ in terms of diagrams. For each $n\ge 1$ we determine presentations for both, ${\mathrm{\text{VTL}}}_n(R^f)$ and ${\mathrm{\text{ATL}}}_n(R^a)$. We determine sequences of homomorphisms ${\{{\rho }_n^f:R^f[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{VTL}}}_n(R^f)\}}_{n=1}^{\infty }$ and ${\{{\rho }_n^a:R^a[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{ATL}}}_n(R^a)\}}_{n=1}^{\infty }$, we determine Markov traces ${\{T_n^{\prime f}:{\mathrm{\text{VTL}}}_n(R^f)\to R^f\}}_{n=1}^{\infty }$ and ${\{T_n^{\prime a}:{\mathrm{\text{ATL}}}_n(R^a)\to R^a\}}_{n=1}^{\infty }$, 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 ${\mathrm{\text{TL}}}_n$ embeds into both, ${\mathrm{\text{VTL}}}_n(R^f)$ and ${\mathrm{\text{ATL}}}_n(R^a)$, and that the restrictions to ${\{{\mathrm{\text{TL}}}_n\}}_{n=1}^{\infty }$ of the two Markov traces coincide.

让 ${\mathrm{电阻}}^F=\mathbb{Z}[{\mathrm{一种}}^{\pm 1}]$ 是变量中洛朗多项式的代数 $\mathrm{一种}$ 然后让 ${\mathrm{电阻}}^{\mathrm{一种}}=\mathbb{Z}[{\mathrm{一种}}^{\pm 1},z_1,z_2,\dots ]$ 是变量中洛朗多项式的代数 $\mathrm{一种}$ 和变量中的标准多项式 $z_1,z_2,\dots .$ 为了 $n\ge 1$ 我们表示为 ${\mathrm{\text{VB}}}_n$ 虚拟编织组 $n$股。我们定义了两个代数塔${\{{\mathrm{\text{VTL}}}_n({\mathrm{电阻}}^F)\}}_{n=1}^{\infty }$ 和 ${\{{\mathrm{\text{ATL}}}_n({\mathrm{电阻}}^{\mathrm{一种}})\}}_{n=1}^{\infty }$在图表方面。对于每个$n\ge 1$ 我们为两者确定演示文稿， ${\mathrm{\text{VTL}}}_n({\mathrm{电阻}}^F)$ 和 ${\mathrm{\text{ATL}}}_n({\mathrm{电阻}}^{\mathrm{一种}})$. 我们确定同态序列${\{{\rho }_n^F：{\mathrm{电阻}}^F[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{VTL}}}_n({\mathrm{电阻}}^F)\}}_{n=1}^{\infty }$ 和 ${\{{\rho }_n^{\mathrm{一种}}：{\mathrm{电阻}}^{\mathrm{一种}}[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{ATL}}}_n({\mathrm{电阻}}^{\mathrm{一种}})\}}_{n=1}^{\infty }$，我们确定马尔可夫迹 ${\{{吨}_n^{\prime F}：{\mathrm{\text{VTL}}}_n({\mathrm{电阻}}^F)\to {\mathrm{电阻}}^F\}}_{n=1}^{\infty }$ 和 ${\{{吨}_n^{\prime \mathrm{一种}}：{\mathrm{\text{ATL}}}_n({\mathrm{电阻}}^{\mathrm{一种}})\to {\mathrm{电阻}}^{\mathrm{一种}}\}}_{n=1}^{\infty }$，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$n\ge 1,$ 标准 Temperley-Lieb 代数 ${\mathrm{\text{TL}}}_n$ 嵌入两者， ${\mathrm{\text{VTL}}}_n({\mathrm{电阻}}^F)$ 和 ${\mathrm{\text{ATL}}}_n({\mathrm{电阻}}^{\mathrm{一种}})$，并且限制 ${\{{\mathrm{\text{TL}}}_n\}}_{n=1}^{\infty }$ 两条马尔可夫迹重合。