Donaldson-Thomas invariants, torus knots, and lattice paths

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 formulas 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 Schröder paths.

Figure 1

Trefoil knot and the corresponding quiver.

Figure 2

A lattice path under the line $y=\frac{1}{4}x$, and a shaded area between the path and the line.

Figure 3

Counting of paths under the line $y=\frac{1}{2}x$ is equivalent to counting excursions, i.e. paths in the upper half plane, made of elementary steps (1,1) and $(1,-2)$.

Figure 4

An example of a bridge (an unconstrained analog of an excursion) made of the same elementary steps as the excursion in Fig. 3.

Figure 5

Coefficients $b_{i,j}^{(2,3)}$ that refine enumeration of paths under the line of the slope $2/3$. Diagonal combinations of these numbers reproduce coefficients in (4.15).

Figure 6

An example of a Schröder path of length 6.

Figure 7

All 6 Schröder paths represented by quadratic terms $(q^2+q^4)x_1^2+(2q+q^3)x_1{x}_2+x_2^2$ of the generating function (6.6).