Congruence Skein Relations for Colored HOMFLY-PT Invariants

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-Marino-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.