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New structures for colored HOMFLY-PT invariants

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Abstract

In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then we apply our results to refine the mathematical structures appearing in the Labastida-Mariño-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another application of the strong integrality, we obtain that the q = 1 and a = 1 specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. We find that a conjectural formula for the colored Alexander polynomial which is the a = 1 specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for framed knots holds.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 12061014).

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  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Shengmao Zhu

  2. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Shengmao Zhu

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Zhu, S. New structures for colored HOMFLY-PT invariants. Sci. China Math. 66, 341–392 (2023). https://doi.org/10.1007/s11425-021-1951-7

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