  We analyze relations between BPS degeneracies related to
Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated
to knots. We introduce a new class of such curves that we call extremal
A-polynomials, discuss their special properties, and determine exact and
asymptotic formulas for the corresponding (extremal) BPS degeneracies. These
formulas lead to nontrivial integrality statements in number theory, as well as
to an improved integrality conjecture stronger than the known M-theory
integrality predictions. Furthermore we determine the BPS degeneracies encoded
in augmentation polynomials and show their consistency with known colored
HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots,
determine them from the knowledge of super-A-polynomials, and verify their
integrality. We illustrate our results with twist knots, torus knots, and
various other knots with up to 10 crossings.
