标题： 环面扭结和广义的施罗德路径 显示英文标题

标题： Torus knots and generalized Schröder paths

摘要： 我们将环面扭结的不变量与一类称为广义Schröder路径的格路计数相关联。 我们确定了位于由所考虑的环面扭结类型决定的区域内的此类路径的生成函数，并证明它们编码了该扭结的着色HOMFLY-PT多项式。 未着色HOMFLY-PT同调的生成元对应于此类路径的基本集合。 通过引入扭结-箭图对应关系，我们将此类路径的生成函数表示为箭图生成级数，并将其与四重分级扭结同调相关联。 此外，我们确定了相应的A多项式，这些多项式为广义Schröder路径的生成函数提供了代数方程和递推关系。 我们感兴趣的格路明确地通过膜构造枚举了与扭结相关的BPS状态。 显示英文摘要

摘要： We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schr\"oder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schr\"oder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions.