  In this paper we find and explore the correspondence between quivers, torus
knots, and combinatorics of counting paths. Our first result pertains to quiver
representation theory -- we find explicit formulae for classical generating
functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We
then focus on quivers corresponding to $(r,s)$ torus knots and show that their
classical generating functions, in the extremal limit and framing $rs$, are
generating 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 Duchon
grammar. Moreover, these extremal A-polynomial equations encode
Donaldson-Thomas invariants, which provides an interesting example of
algebraicity of generating functions of these invariants. We also find a
quantum generalization of these statements, i.e. a relation between motivic
quiver generating functions, quantum extremal knot invariants, and $q$-weighted
path counting. Finally, in the case of the unknot, we generalize this
correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder
paths.
