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In this paper we find and explore the correspondence between quivers, torusknots, and combinatorics of counting paths. Our first result pertains to quiverrepresentation theory -- we find explicit formulae for classical generatingfunctions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. Wethen focus on quivers corresponding to $(r,s)$ torus knots and show that theirclassical generating functions, in the extremal limit and framing $rs$, aregenerating functions of lattice paths under the line of the slope $r/s$.Generating functions of such paths satisfy extremal A-polynomial equations,which immediately follows after representing them in terms of the Duchongrammar. Moreover, these extremal A-polynomial equations encodeDonaldson-Thomas invariants, which provides an interesting example ofalgebraicity of generating functions of these invariants. We also find aquantum generalization of these statements, i.e. a relation between motivicquiver generating functions, quantum extremal knot invariants, and $q$-weightedpath counting. Finally, in the case of the unknot, we generalize thiscorrespondence to the full HOMFLY-PT invariants and counting of Schr\"oderpaths.

Donaldson-Thomas invariants, torus knots, and lattice paths