论文标题
Homflypt镜头空间的Skein子模块$ L（P，1）$
HOMFLYPT skein sub-modules of the lens spaces $L(p, 1)$
论文作者
论文摘要
在本文中，我们通过辫子向$ l（p，1）$，$ \ mathcal {s}（l（p，1））$的$ l（p，1）$的Homflypt绞线模块努力。我们的出发点是固体圆环st的Homflypt skein模块的线性turaev-basis，$λ^{\ prime} $，$ \ mathcal {s}（{\ rm st}）$，可以分解为“阳性” $ {λ^prime}的张量， $ {λ^{\ prime}}^ - $ sub-modules和lambropoulou不变式，$ x $，用于ST中的结和链接，可捕获$ s（{\ rm st}）$。到目前为止，$ \ MATHCAL {s}（l（p，1））= \ frac {\ Mathcal {s}（st）} {<bbm's>} $，其中BBM的（BRAID BAND MOVES）表示与与$ L（pergery of $ l（p，1）$相对应。也就是说，通过强加从BBM的表现并解决以这种方式获得的无限方程式的关系，可以将ST中的Homflypt型不变和链接扩展到$ L（p，1）$中的结（p，1）$中的不变性和链接。 \ Small Break在本文中，我们以$ \ Mathcal {s}（{\ rm st}）$，$λ$的新基础合作，我们将通过在$λ^+$中执行$λ^+$中的元素来获得的无限方程式与$ i $ i $ i $ i $ i $ i $ i $ i $ i $ i.更确切地说，我们证明一个系统的解决方案可以从另一个系统的解决方案中得出。我们的目的是降低无限系统的复杂性，以便使用辫子技术来计算$ \ mathcal {s}（l（p，1））$。最后，我们提供了一个生成集和$ \ frac {λ^+} {<bbm's>} $的潜在基础，因此，我们获得了$ \ frac {λ^ - } {<bbm's>} $的生成集和潜在基础。我们还讨论了通过辫子计算$ \ Mathcal {s}（l（p，1））$所需的进一步步骤。
In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $Λ^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${Λ^{\prime}}^+$ and the "negative" ${Λ^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{<bbm's>}$, where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of $L(p, 1)$. Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in $L(p, 1)$ by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. \smallbreak In this paper we work with a new basis of $\mathcal{S}({\rm ST})$, $Λ$, and we relate the infinite system of equations obtained by performing bbm's on elements in $Λ^+$ to the infinite system of equations obtained by performing bbm's on elements in $Λ^-$ via a map $I$. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute $\mathcal{S}(L(p,1))$ using the braid technique. Finally, we present a generating set and a potential basis for $\frac{Λ^+}{<bbm's>}$ and thus, we obtain a generating set and a potential basis for $\frac{Λ^-}{<bbm's>}$. We also discuss further steps needed in order to compute $\mathcal{S}(L(p,1))$ via braids.
