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)
Pages (from-to)683-729
Number of pages47
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jun 2023

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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