Incompressibility criteria for spun-normal surfaces

TL;DR

This paper introduces a simple criterion for identifying incompressible spun-normal surfaces in ideal triangulations, leading to new results in knot theory and the proof of the Slope Conjecture for certain 2-fusion knots.

Contribution

It provides a practical incompressibility criterion based on normal surface theory, connecting topological properties with character variety techniques.

Findings

Existence of alternating knots with non-integer boundary slopes

Proof of the Slope Conjecture for a large class of 2-fusion knots

Example of spun-normal surfaces not arising from ideal points of the deformation variety

Abstract

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the existence of alternating knots with non-integer boundary slopes, and a proof of the Slope Conjecture for a large class of 2-fusion knots. While the condition and conclusion are purely topological, the proof uses the Culler-Shalen theory of essential surfaces arising from ideal points of the character variety, as reinterpreted by Thurston and Yoshida. The criterion itself comes from the work of Kabaya, which we place into the language of normal surface theory. This allows the criterion to be easily applied, and gives the framework for proving that the surface is incompressible. We also explore which spun-normal surfaces arise from ideal points of the deformation variety. In particular, we give an example where no vertex or fundamental surface arises in this way.