A knot without a nonorientable essential spanning surface

TL;DR

This paper presents the first example of a hyperbolic knot in the 3-sphere lacking a nonorientable essential spanning surface, challenging existing conjectures about boundary slopes and surface existence.

Contribution

It provides a counterexample to the Strong Neuwirth Conjecture and the Even Boundary Slope Conjecture using Thurston's spun-normal surfaces and normal surface algorithms.

Findings

Counterexample to the Strong Neuwirth Conjecture

Counterexample to the Even Boundary Slope Conjecture

Rigorous calculation using Thurston's spun-normal surfaces

Abstract

This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the same authors. The proof is a rigorous calculation using Thurston's spun-normal surfaces in the spirit of Haken's original normal surface algorithms.