## Abstract

The original HOMFLY-PT polynomials can be fully determined by a very simple rule, the skein relation, while the colored HOMFLY-PT invariants (2 variables) of

links are notoriously hard to compute. Inspired by the large N duality connecting Chern–Simons gauge theory and topological string theory, the Labastida-Mariño–Ooguri–Vafa (LMOV) conjecture for links (or framed links) predicts integrality, pole order structure and symmetric property for the colored HOMFLY-PT invariants. By studying the LMOV conjecture for framed links, we uncover certain congruence skein relations for the (reformulated) colored HOMFLY-PT invariants. Although these congruence skein relations still can not fully determine the colored HOMFLY-PT invariants, they provide a strong pattern for the colored HOMFLY-PT invariants, which possibly could pave a way for people to understand the very mysterious nature of the colored HOMFLY-PT invariants. We prove that these congruence skein relations hold in many different situations.

Finally, we discuss the applications of the congruence skein relations in framed

LMOV conjecture.

links are notoriously hard to compute. Inspired by the large N duality connecting Chern–Simons gauge theory and topological string theory, the Labastida-Mariño–Ooguri–Vafa (LMOV) conjecture for links (or framed links) predicts integrality, pole order structure and symmetric property for the colored HOMFLY-PT invariants. By studying the LMOV conjecture for framed links, we uncover certain congruence skein relations for the (reformulated) colored HOMFLY-PT invariants. Although these congruence skein relations still can not fully determine the colored HOMFLY-PT invariants, they provide a strong pattern for the colored HOMFLY-PT invariants, which possibly could pave a way for people to understand the very mysterious nature of the colored HOMFLY-PT invariants. We prove that these congruence skein relations hold in many different situations.

Finally, we discuss the applications of the congruence skein relations in framed

LMOV conjecture.

Original language | English |
---|---|

Pages (from-to) | 683-729 |

Number of pages | 47 |

Journal | Communications in Mathematical Physics |

Volume | 400 |

Issue number | 2 |

Publication status | Published - Jun 2023 |