Torus decomposition and foliation detected slopes

TL;DR

This paper proves that if a closed 3-manifold formed by gluing two knot manifolds admits a co-oriented taut foliation, then the gluing map aligns certain boundary slopes detected by the foliation, confirming a conjecture.

Contribution

It establishes a link between taut foliations and boundary slopes in glued knot manifolds, confirming a conjecture by Boyer, Gordon, and Hu.

Findings

If the manifold admits a taut foliation, the gluing map matches CTF-detected slopes.

The result confirms the conjecture relating foliation detection and boundary slopes.

Provides new insights into the structure of 3-manifolds with taut foliations.

Abstract

Let $M_1$ and $M_2$ be knot manifolds and $M=M_1\cup_f M_2$ be the closed 3-manifold obtained by gluing up $M_1$ and $M_2$ via $f:\partial M_1\xrightarrow{\cong} \partial M_2$. We show that if $M$ admits a co-oriented taut foliation, then $f$ identifies some CTF-detected rational boundary slopes of $M_1$ and $M_2$, affirming a conjecture proposed by Boyer, Gordon and Hu.