| Résume||[Séminaire Commun Paris7-Paris13->www.math.univ-paris13.fr/laga/index.php/fr/ta/seminaires]
In this talk I will present some of the aspects of the categorification of the colored HOMFLY-PT polynomials for knots and links. A rigorous categorification of these two-variable polynomials and their specialization exist only for certain colors/representations. Apart from a brief overview of these constructions, I'll present a list of conjectural structural properties that these homology theories should satisfy, motivated by physics-inspired relationship with the BPS homology. In turn, they enable new insights on the colored HOMFLY-PT polynomials, the recursion relations they satisfy, as well as the explicit computation of the corresponding BPS numbers, giving new and surprising integrality properties.