  We introduce and explore the relation between knot invariants and quiver
representation theory, which follows from the identification of quiver quantum
mechanics in D-brane systems representing knots. We identify various structural
properties of quivers associated to knots, and identify such quivers explicitly
in many examples, including some infinite families of knots, all knots up to 6
crossings, and some knots with thick homology. Moreover, based on these
properties, we derive previously unknown expressions for colored HOMFLY-PT
polynomials and superpolynomials for various knots. For all knots, for which we
identify the corresponding quivers, the LMOV conjecture for all symmetric
representations (i.e. integrality of relevant BPS numbers) is automatically
proved.
