Quantum Invariants and Fiberedness

TL;DR

This paper investigates the topological significance of the Gukov-Manolescu knot series, linking it to invariants like the Hopf invariant, colored Jones polynomials, and boundary slopes, with implications for fiberedness and knot homology.

Contribution

It provides explicit formulas connecting the Gukov-Manolescu series to classical invariants and proposes a slope conjecture relating it to boundary slopes.

Findings

Leading coefficient of $F_K$ is a monomial for certain knots.

Explicit formula for Hopf invariant in terms of colored Jones polynomials.

Proposes a slope conjecture linking $F_K$ to boundary slopes.

Abstract

We explore the topological significance of the Gukov-Manolescu knot series $F_K$. We show that the leading coefficient of $F_K$ is a monomial and express its exponent in terms of the Hopf invariant for all homogeneous braid knots, and for fibered knots up to 12 crossings. As an application, we deduce an explicit formula for the Hopf invariant in terms of colored Jones polynomials. For non-fibered strongly quasipositive knots, we study a relation between $F_K$ and the stability series of the colored Jones function, and explore similarities between $F_K$ and knot Floer homology. Finally, we propose a slope conjecture for $F_K$, relating it to the boundary slopes of the knot.