The knot meridians of (1,1)-knot complements are CTF-detected

TL;DR

This paper constructs specific taut foliations in the complements of non-simple (1,1)-knots, providing evidence for a conjecture linking foliation detection to the L-space conjecture.

Contribution

It introduces a method to detect knot meridians using co-oriented taut foliations in (1,1)-knot complements, advancing understanding of the L-space conjecture.

Findings

Constructed taut foliations intersecting boundary transversely

Provides evidence linking foliation detection to the L-space conjecture

Supports the conjecture by Boyer, Gordon, and Hu

Abstract

For any non-simple (1,1)-knot in $S^3$ or a lens space, we construct a co-oriented taut foliation in its complement that intersects the boundary torus transversely in a suspension foliation of the knot meridian, or the infinity slope. This provides new evidence for a conjecture made by Boyer, Gordon and Hu using slope detections, related to the L-space conjecture.