Density spectra for knots

TL;DR

This paper explores the relationship between various density spectra of knots, extending previous work to include quantum invariants like Jones polynomials and knot homology, and proposes new conjectures.

Contribution

It introduces quantum density spectra for knots and formulates related conjectures, expanding the understanding of knot invariants beyond classical volume and determinant spectra.

Findings

Established connections between quantum and classical density spectra.

Proposed conjectures relating quantum invariants to geometric properties.

Extended the spectrum analysis to include Jones polynomials and knot homology.

Abstract

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.