Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Congruent skein relations for colored HOMFLY-PT invariants and colored
  Jones polynomials   Colored HOMFLY-PT invariant, the generalization of the colored Jones
polynomial, is one of the most important quantum invariants of links. This
paper is devoted to investigating the basic structures of the colored HOMFLY-PT
invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that
the (reformulated) colored HOMFLY-PT invariants actually lie in the ring
$\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric
formulas for colored HOMFLY-PT invariants of links, which include the
rank-level duality as an easy consequence. Finally, motivated by the
Labastida-Mari\~no-Ooguri-Vafa conjecture for framed links, we propose
congruent skein relations for (reformulated) colored HOMFLY-PT invariants which
are the generalizations of the skein relation for classical HOMFLY-PT
polynomials. Then we study the congruent skein relation for colored Jones
polynomials. In fact, we obtain a succinct formula for the case of knot. As an
application, we prove a vanishing result for Reshetikhin-Turaev invariants of a
family of 3-manifolds. Finally we study the congruent skein relations for
$SU(n)$ quantum invariants.

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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph