Double twist knots and lattice paths

TL;DR

This paper investigates the combinatorial structures of twist and double twist knots through quiver generating series of their HOMFLY-PT polynomials, linking knot invariants to lattice path models.

Contribution

It introduces a novel connection between knot invariants and lattice path models via quiver generating series for specific classes of knots.

Findings

Lattice path models derived from the HOMFLY-PT polynomial for twist and double twist knots.

Limit cases of the polynomial relate to combinatorial lattice path structures.

Provides a new combinatorial perspective on knot invariants.

Abstract

In this work, we explore the combinatorics arising from the quiver generating series of the unreduced $r$-colored HOMFLY-PT polynomial $\bar{P}_r(a,q)$ for some twist-knots and double twist knots. By taking the limit $a = 0$ and $q = 1$, we indeed obtain lattice path models for these knots.