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The colored HOMFLY polynomial is a quantum invariant of oriented links in S³ associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure, which is the part of Labastida-Marino-Ooguri-Vafa conjecture. The new integer invariants are also predicted by the LMOV conjecture and recently has been proved. LMOV also give the application of Licherish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial could also be developed in a same fashion by using more complicated algebra method.
In a joint work with Lin Chen and Nicolai Reshetikhin, we rigorously formulate the orthogonal quantum group version of LMOV conjecture in mathematics by using the representation of Brauer centralizer algebra. We also obtain formulae of Lichorish-Millet type which could be viewed as the application in knot theory and topology. By using the cabling technique, we obtain a uniform formula of colored Kauffman polynomial for all torus links with all partitions. Combined these together, we are able to prove many interesting cases of this orthogonal LMOV conjecture.