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

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