Quiver Diagonalization and Open BPS States

We show that motivic Donaldson-Thomas invariants of a~symmetric quiver $Q$, captured by the generating function $P_Q$, can be encoded in another $Q^{(\infty)}$ (almost always) infinite size, whose only arrows are loops, and $P_{Q^{(\infty)}}$ is equal to $P_Q$ upon appropriate identification parameters. Consequences this statement include a generalization proof integrality Labastida-Mari\~{n}o-Ooguri-Vafa count open BPS states, as well expressing an arbitrary symmetric terms $m$-loop quivers. In particular, means already known combinatorial interpretation quivers extends