  We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory
with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we
provide the explicit form of low-energy effective actions of Type IIA string
theory with D4-branes on the $\Omega$-background. This form enables us to
relate refined Chern-Simons invariants of a torus knot/link in $S^3$ to refined
BPS invariants in the resolved conifold. Assuming that the extra
$\mathrm{U}(1)$ global symmetry acts on BPS states trivially, the duality
predicts graded dimensions of cohomology groups of moduli spaces of M2-M5 bound
states associated to a torus knot/link in the resolved conifold. Thus, this
formulation can be interpreted as a positivity conjecture of refined
Chern-Simons invariants of torus knots/links. We also discuss about an
extension to non-torus knots.
