The signature and cusp geometry of hyperbolic knots
This work addresses a problem in knot theory by providing new inequalities that connect geometric and topological invariants, with potential applications in low-dimensional topology.
The authors tackled the problem of relating hyperbolic knot invariants by introducing a new invariant called the natural slope, derived from cusp geometry, and proved 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, with applications to Dehn surgery and 4-ball genus.
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.