We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.

中文翻译：

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Knots-quivers 对应

我们介绍并探索了结不变量与颤动表示理论之间的关系，这源于在表示结的 D-膜系统中颤动量子力学的识别。我们确定了与结相关的箭袋的各种结构特性，并在许多例子中明确地确定了这样的箭袋，包括一些无限的结族，所有结多达 6 个交叉点，以及一些具有厚同源性的结。此外，基于这些特性，我们为各种结的彩色 HOMFLY-PT 多项式和超多项式推导出了以前未知的表达式。对于我们确定相应箭袋的所有结，所有对称表示（即相关 BPS 数的完整性）的 LMOV 猜想都会自动证明。