TY - JOUR
T1 - Full colored HOMFLYPT invariants, composite invariants and congruence skein relations
AU - Chen, Qingtao
AU - Zhu, Shengmao
N1 - Funding Information:
The authors appreciate the collaboration with Kefeng Liu and Pan Peng in this area and many valuable discussions with them within the past years. The authors also thank Rinat Kashaev, Jun Murakami and Nicolai Reshetikhin for their interests, encouragements and discussions. The authors thank the Shanghai Mathematical Center for its support and hospitality, where the partial work was finished. 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.
Funding Information:
The authors appreciate the collaboration with Kefeng Liu and Pan Peng in this area and many valuable discussions with them within the past years. The authors also thank Rinat Kashaev, Jun Murakami and Nicolai Reshetikhin for their interests, encouragements and discussions. The authors thank the Shanghai Mathematical Center for its support and hospitality, where the partial work was finished. 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:
© 2020, Springer Nature B.V.
PY - 2020/12
Y1 - 2020/12
N2 - In this paper, we investigate the properties of certain quantum invariants of links by using the HOMFLY skein theory. First, we obtain the limit behavior for the full colored HOMFLYPT invariant which is the natural generalization of the colored HOMFLYPT invariant. Then we focus on the composite invariant which is a certain combination of the full colored HOMFLYPT invariants. Motivated by the study of the Labastida–Mariño–Ooguri–Vafa conjecture for the framed composite invariants of links, we introduce the notion of reformulated composite invariant Rˇ p(L; q, a). By using the HOMFLY skein theory, we prove that Rˇ p(L; q, a) actually lies in the integral ring 2Z[(q-q-1)2,a±1]. Finally, we propose a conjectural congruence skein relation for Rˇ p(L; q, a) and prove it for certain special cases.
AB - In this paper, we investigate the properties of certain quantum invariants of links by using the HOMFLY skein theory. First, we obtain the limit behavior for the full colored HOMFLYPT invariant which is the natural generalization of the colored HOMFLYPT invariant. Then we focus on the composite invariant which is a certain combination of the full colored HOMFLYPT invariants. Motivated by the study of the Labastida–Mariño–Ooguri–Vafa conjecture for the framed composite invariants of links, we introduce the notion of reformulated composite invariant Rˇ p(L; q, a). By using the HOMFLY skein theory, we prove that Rˇ p(L; q, a) actually lies in the integral ring 2Z[(q-q-1)2,a±1]. Finally, we propose a conjectural congruence skein relation for Rˇ p(L; q, a) and prove it for certain special cases.
KW - Colored HOMFLYPT invariants
KW - Composite invariants
KW - Congruence skein relations
KW - HOMFLY skein theory
KW - LMOV conjecture
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U2 - 10.1007/s11005-020-01327-4
DO - 10.1007/s11005-020-01327-4
M3 - Article
AN - SCOPUS:85090311653
SN - 0377-9017
VL - 110
SP - 3307
EP - 3342
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
IS - 12
ER -