Flow loops and quantum groups

TL;DR

This paper introduces a new dynamical invariant for fibered knots based on Morse flow loops, conjecturally matching quantum group invariants and verified for braid-homogeneous knots.

Contribution

It defines a two-variable series invariant from Morse flows and proves its equivalence to the BPS q-series for all braid-homogeneous knots.

Findings

Conjectural equivalence between the dynamical series and BPS q-series.

Proof of this correspondence for all braid-homogeneous knots.

Establishes a link between Morse flow dynamics and quantum group invariants.

Abstract

This paper connects two seemingly different ways of studying knots: quantum group invariants and the dynamics of Morse flows. For fibered knots, we define a two-variable series invariant by counting Morse flow loops in the complement. This dynamical series is conjectured to agree with the BPS $q$-series of the knot complement, which arises from Verma modules for quantum groups and encodes all colored Jones polynomials. We prove this correspondence for all braid-homogeneous knots.