New structures for colored HOMFLY-PT invariants (Q2683043)

In the paper under review, the author describes new structures for the colored HOMFLY-PT invariant of framed links. In particular, the strong integrality property is proved by using the HOMFLY-PT skein theory developed by Morton and his collaborators. From this, several symmetric properties for the full colored HOMFLY-PT invariants of links are derived. As a consequence of this analysis, the mathematical structures appearing in the Labastida-Marino-Ooguri-Vafa (LMOV) integrality conjecture for framed links is refined. The strong integrality also allows the author to obtain that the \(q=1\) and \(a=1\) specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. A conjectural formula for the colored Alexander polynomial is also shown to imply a special case of the LMOV conjecture for framed knots.