论文标题
镜头空间的Kauffman支架旋转模块通过无调的辫子
The Kauffman bracket skein module of the lens spaces via unoriented braids
论文作者
论文摘要
在本文中，我们开发了一种辫子理论方法，用于计算镜头空间$ l（p，q）$，kbsm（$ l（p，q）$）的kauffman支架绞线模块，以$ q \ neq 0 $。为此，我们介绍了一个新概念，该概念是{\ it无定向的编织}。通过忽略链的自然到底的自然方向，从标准辫子中获得了无定向的辫子。 We first define the {\it generalized Temperley-Lieb algebra of type B}, $TL_{1, n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on $TL_{1, n}$.通用不变$ V $等于KBSM（ST）。对于现在传递到KBSM（$ L（p，q）$），我们强加于$ v $关系来自乐队移动（或幻灯片移动），即反映$ L（p，q）$但不在ST中的移动，而不是在ST中，并且反映了对$ L（P，Q）$的手术描述，因此获得了一个不适式系统的均等系统。通过构造，求解这个无限的方程系统等于计算KBSM（$ L（p，q）$）。我们首先提出解决方案$ q = 1 $，这对应于获得新基础，$ \ nathcal {b} _ {p} $，用于KBSM（$ l（p，1）$），并使用$（\ lfloor p/2 \ rfloor +1）$ elements。我们注意到，基础$ \ MATHCAL {B} _ {P} $与Hoste \＆przytycki获得的基础不同。为了处理CASE $ Q> 1 $的无限系统的复杂性，我们首先展示了如何使用基于无调编织的图形方法来获得新的基础$ \ MathCal {b} _ {p} $ kbsm（$ l（p，1）$），我们最终将结果扩展到了我们的结果，我们最终将结果扩展到case $ q> 1 $ q> 1 $ q> 1 $ q> 1 $ q>> 1 $ q> 1 $ q>> 1 $。
In this paper we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\neq 0$. For doing this, we introduce a new concept, that of an {\it unoriented braid}. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the {\it generalized Temperley-Lieb algebra of type B}, $TL_{1, n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on $TL_{1, n}$. The universal invariant $V$ is equivalent to the KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description of $L(p,q)$, obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM($L(p,q)$). We first present the solution for the case $q=1$, which corresponds to obtaining a new basis, $\mathcal{B}_{p}$, for KBSM($L(p,1)$) with $(\lfloor p/2 \rfloor +1)$ elements. We note that the basis $\mathcal{B}_{p}$ is different from the one obtained by Hoste \& Przytycki. For dealing with the complexity of the infinite system for the case $q>1$, we first show how the new basis $\mathcal{B}_{p}$ of KBSM($L(p,1)$) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case $q>1$.
