TY - JOUR
T1 - Congruence Skein Relations for Colored HOMFLY -PT Invariants
AU - Chen, Qingtao
AU - Liu, Kefeng
AU - Peng, Pan
AU - Zhu, Shengmao
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/6
Y1 - 2023/6
N2 - 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.
AB - 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.
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U2 - 10.1007/s00220-022-04604-6
DO - 10.1007/s00220-022-04604-6
M3 - Article
AN - SCOPUS:85143796998
SN - 0010-3616
VL - 400
SP - 683
EP - 729
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -