Note on Euler class 0 taut foliation on the Whitehead link exterior

TL;DR

This paper investigates the existence of co-orientable taut foliations on the Whitehead link exterior, revealing significant obstructions related to Euler classes and contrasting with other knot exteriors, using advanced topological techniques.

Contribution

It introduces new obstructions to taut foliations with certain Euler classes on the Whitehead link exterior, expanding understanding of foliation existence criteria in hyperbolic 3-manifolds.

Findings

Most lattice points in the Thurston ball cannot be realized as Euler classes

Limited exceptional cases exist for realizability

Develops general methods for studying foliations on cusped hyperbolic 3-manifolds

Abstract

This paper studies the existence of co-orientable taut foliations on 3-manifolds, particularly focusing on the Whitehead link exterior. We demonstrate fundamental obstructions to the existence of such foliations with certain Euler class properties, contrasting with the more permissive case of the figure-eight knot exterior. Our analysis reveals that most lattice points in the dual unit Thurston ball of the Whitehead link exterior cannot be realized as Euler classes of co-orientable taut foliations, with only limited exceptional cases. The proofs combine techniques from foliation theory, including saddle fillings and index formulas, with topological obstructions derived from Dehn surgery. These results yield general methods for studying foliations on cusped hyperbolic 3-manifolds. These techniques applied to broader classes of manifolds beyond the specific examples considered here