Transverse tori in Engel manifolds

Abstract

We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds: Every torus with trivial normal bundle is isotopic to infinitely many distinct transverse tori, distinguished locally (and globally in the nullhomologous case) by their formal invariants. (Few examples of transverse tori were previously known.) We classify the formal invariants, which are richer than for transverse knots. We show that in an overtwisted Engel structure, up to homotopy through such structures, these invariants are a complete set of uniqueness obstructions, and every torus with trivial normal bundle can be made transverse realizing any combination of these invariants. Fixing Engel structures not known to be overtwisted, we explore the range of the primary invariants of given tori. A sample application is that many Engel manifolds admit infinitely many transverse homotopy classes of unknotted transverse tori such that each class contains infinitely many transverse isotopy classes.