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Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
Communications in Number Theory and Physics ( IF 1.9 ) Pub Date : 2019-01-01 , DOI: 10.4310/cntp.2019.v13.n4.a3
Petr Dunin-Barkowski 1 , Aleksandr Popolitov 2 , Sergey Shadrin 3 , Alexey Sleptsov 4
Affiliation  

We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mari\~no spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data.

中文翻译:

环结的彩色 HOMFLY-PT 多项式的组合结构

我们根据自由费米子(半无限楔形)形式为环面结的彩色 HOMFLY-PT 多项式重写(扩展)Ooguri-Vafa 分区函数,使其与双 Hurwitz 数的生成函数非常相似。这使我们能够推测通过在 Brini-Eynard-Mari\~no 光谱曲线上的拓扑递归获得的相关微分的完全展开的组合含义,用于环结的彩色 HOMFLY-PT 多项式。这种对应关系表明扩展的 Ooguri-Vafa 分配函数的结构组合结果。即,其系数应具有拟多项式行为,其中非多项式因子由雅可比多项式给出。我们以纯粹的组合方式证明了这个拟多项式。除此之外,我们还表明 (0,1)- 和 (0,
更新日期:2019-01-01
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