The signature and cusp geometry of hyperbolic knots

We introduce a new real-valued invariant called the natural slope of a
hyperbolic knot in the 3-sphere, which is defined in terms of its cusp
geometry. We show that twice the knot signature and the natural slope differ by
at most a constant times the hyperbolic volume divided by the cube of the
injectivity radius. This inequality was discovered using machine learning to
detect relationships between various knot invariants. It has applications to
Dehn surgery and to 4-ball genus. We also show a refined version of the
inequality where the upper bound is a linear function of the volume, and the
slope is corrected by terms corresponding to short geodesics that link the knot
an odd number of times.

Authors

Alex Davies, András Juhász, Marc Lackenby, Nenad Tomasev