2014/02/14 by Qingtao Chen, Chen, Qingtao, Kefeng Liu +5
Mathematics · Physics and Astronomy · #57M25 #57M27 #81R50 #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #hep-th #math-ph #math.GT #math.MP #math.QA #math.RT #msc:57M25 #msc:57M27 #msc:81R50
paper · pdf · doi:10.48550/arxiv.1402.3571
50 pages, 7 figures. More results are added and we rewrite the paper in a succinct way
arxiv created 2015/11/16 · arxiv updated 2015/11/17
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 ℤ[(q-q-1)2,t± 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-Mariño-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.