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Quantum Invariants and Fiberedness

Paul Orland, Lara San Martín Suárez, Toby Saunders-A'Court, Josef Svoboda

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.

Quantum Invariants and Fiberedness

Abstract

We explore the topological significance of the Gukov-Manolescu knot series . We show that the leading coefficient of 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 and the stability series of the colored Jones function, and explore similarities between and knot Floer homology. Finally, we propose a slope conjecture for , relating it to the boundary slopes of the knot.
Paper Structure (39 sections, 13 theorems, 133 equations, 7 figures, 4 tables)

This paper contains 39 sections, 13 theorems, 133 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. GM series and colored Jones function
  3. Main results
  4. Non-fibered knots
  5. Relation to knot Floer homology
  6. Slopes
  7. Conventions
  8. Preliminaries
  9. Inverted state sum
  10. Inversion data
  11. $R$-matrices
  12. Valid states
  13. State sum
  14. The leading term
  15. The leading term and the ground state
  16. ...and 24 more sections

Key Result

Proposition 1

Let $K$ be a nice knot with a choice of braid diagram $\beta$ and inversion datum $\iota$. Then, the ground state is the only state that contributes to the leading term of the inverted state sum.

Figures (7)

  • Figure 1: Inversion datum on a braid representative of $12n_{423}$.
  • Figure 2: The figure-eight knot $4_1$ represented as a closure of the braid $\beta = \sigma_2 \sigma_1^{-1} \sigma_2 \sigma_1^{-1}$.
  • Figure 3: Every crossing $c\in V_\beta$ is connected to four adjacent segments in $E_\beta$, denoted by $e_{BR}$, $e_{BL}$, $e_{TR}$ and $e_{TL}$ (bottom/top, right/left). The components of a state at a crossing $c$ are denoted by $i,j,i',j'$.
  • Figure 5: The braid $\sigma_{u,v}$ and its labels.
  • Figure 6: The double twist links $C_{m,n}$.
  • ...and 2 more figures

Theorems & Definitions (62)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5: Slope conjecture for $F_K$
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: ParkThesis
  • Definition 5
  • ...and 52 more