In this talk I shall present the knots-quivers correspondence, as well as some surprising implications in combinatorics involving counting of lattice paths and number theory. The knots-quivers correspondence relates the colored HOMFLY-PT invariants of a knot with the motivic Donaldson-Thomas invariants of the corresponding quiver. This correspondence is made completely explicit at the level of generating series. The motivation for this relationship comes from topological string theory, BPS (LMOV) invariants, as well as categorification of HOMFLY-PT polynomial and A-polynomials. We compute quivers for various classes of knots, including twist knots, rational knots and torus knots.

One of the surprising outcomes of this correspondence is that from the information of the colored HOMFLY-PT polynomials of certain knots we get new expressions for the classical combinatorial problem of counting lattice paths, as well as new integrality/divisibility properties.

The main goal of this talk is to present basic ideas and to present numerous open questions and ramifications coming from knots-quivers correspondence.

(based on joint works with P. Sulkowski, M. Reineke, P. Kucharski, M. Panfil and P. Wedrich).