Full colored HOMFLYPT invariants, composite invariants and congruence skein relations
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Full colored HOMFLYPT invariants, composite invariants and congruence skein relations Qingtao Chen1 · Shengmao Zhu2 Received: 21 September 2017 / Revised: 22 March 2020 / Accepted: 27 August 2020 © Springer Nature B.V. 2020
Abstract 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 ˇ p (L; q, a). By using the introduce the notion of reformulated composite invariant R ˇ p (L; q, a) actually lies in the integral ring HOMFLY skein theory, we prove that R 2Z[(q − q −1 )2 , a ±1 ]. Finally, we propose a conjectural congruence skein relation for ˇ p (L; q, a) and prove it for certain special cases. R Keywords Colored HOMFLYPT invariants · HOMFLY skein theory · Composite invariants · Congruence skein relations · LMOV conjecture Mathematics Subject Classification 57K14 · 57K16 · 57K31
Contents 1 Introduction . . . . . . . . . . . . 2 HOMFLY skein theory . . . . . . . 2.1 The plane . . . . . . . . . . . 2.2 The rectangle . . . . . . . . . 2.3 The annulus . . . . . . . . . . 2.4 Skein involutions . . . . . . . 3 Full colored HOMFLYPT invariants
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Shengmao Zhu [email protected]
1
Division of Science, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates
2
School of Mathematics and Shing-Tung Yau Center, Southeast University, Nanjing 211189, China
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Q. Chen, S. Zhu 3.1 Partitions and symmetric functions . . . . . . . . . . . . . . . 3.2 Basic elements in C . . . . . . . . . . . . . . . . . . . . . . . 3.3 The meridian maps of C . . . . . . . . . . . . . . . . . . . . . 3.4 Construction of the elements Q λ,μ . . . . . . . . . . . . . . . 3.5 Full colored HOMFLYPT invariants . . . . . . . . . . . . . . 3.6 Symmetric properties . . . . . . . . . . . . . . . . . . . . . . 4 Full colored HOMFLYPT invariants for torus links . . . . . . . . . 5 Special polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 6 Composite invariants and integrality property . . . . . . . . . . . . 6.1 LMOV conjecture for composite invariants . . . . . . . . . . . 6.2 Framed LMOV conjecture for the framed composite invariants 7 Reformulated composite invariant
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