Congruence Skein Relations for Colored HOMFLY -PT Invariants

Qingtao Chen, Kefeng Liu, Pan Peng, Shengmao Zhu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
JournalCommunications In Mathematical Physics
StateAccepted/In press - 2022

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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