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 inje...
| Main Authors: | , , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Mathematical Sciences Publishers
2024
|
| Summary: | 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. |
|---|