AI 中文总结
研究有限和大N时G = SU(N)纽结补的3d-3d对应，通过例子将纽结补同调块F_K实现为3d N = 2理论半指标，可据此实现彩色HOMFLY-PT多项式，还描述了获取同调块及其a变形版本的方法并讨论配分函数性质。
AI 中文摘要
对于有限和大N的G = SU(N)，在完全对称表示下，通过研究一些例子，我们将纽结补S^3 \ K的同调块F_K实现为3d N = 2理论T[M_3]的半指标，其形式为倒转的哈比罗级数，我们期望将其推广到一般纽结。从半指标表达式中，通过取一组特定极点也能实现彩色HOMFLY-PT多项式。通过半指标实现，我们描述了一种从彩色HOMFLY-PT多项式的哈比罗级数表达式中获取S^3 \ K的G = SU(N)同调块及其a变形版本的方法。我们还讨论了任意N的配分函数的一些性质。
英文摘要
For $G=SU(N)$ at finite and large $N$, with a totally symmetric representation, we realize the homological block $F_K$ for a knot complement $S^3 \backslash K$, given in the form of the inverted Habiro series, as a half-index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ by studying some examples, which we expect to extend to general knots. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by taking a certain set of poles. Through the half-index realization, we describe a method for obtaining the $G=SU(N)$ homological block and its $a$-deformed version for $S^3 \backslash K$ from a Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary $N$.
Comments37 pages
