TY - JOUR
T1 - Congruence Skein Relations for Colored HOMFLY -PT Invariants
AU - Chen, Qingtao
AU - Liu, Kefeng
AU - Peng, Pan
AU - Zhu, Shengmao
N1 - Funding Information:
We would like to thank N. Reshetikhin, J. Murakami, E. Witten and Tian Yang for valuable discussions with us. We are very grateful to the anonymous referees for careful reading of our paper and for their valuable comments. Q. Chen thank Dror Bar-Natan and Scott Morrison for communicating on KnotTheory, Package of Mathematica. Both Q. Chen and S. Zhu thank Shanghai Center for Mathematical Science for their hospitality. The research of Q. Chen is supported by the National Centre of Competence in Research SwissMAP of the Swiss National Science Foundation and the start-up research grant at NYU Abu Dhabi. The research of S. Zhu is supported by the National Science Foundation of China grant No. 11201417 and the China Postdoctoral Science special Foundation No. 2013T60583.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022
Y1 - 2022
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.
UR - http://www.scopus.com/inward/record.url?scp=85143796998&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85143796998&partnerID=8YFLogxK
U2 - 10.1007/s00220-022-04604-6
DO - 10.1007/s00220-022-04604-6
M3 - Article
AN - SCOPUS:85143796998
SN - 0010-3616
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
ER -