Some 3-dimensional transverse C-links (Constructions of higher-dimensional C-links, I)

TL;DR

This paper explores the construction of higher-dimensional C-links using techniques involving quasipositive knots and links, demonstrating how many smooth 3-manifolds can be realized as transverse intersections in complex 3-space, with implications for contact structures.

Contribution

It introduces new methods for constructing higher-dimensional C-links and demonstrates how various 3-manifolds can be realized as transverse intersections with complex surfaces.

Findings

Many smooth 3-manifolds are realized as transverse intersections in complex 3-space.

Constructed examples have canonical Stein-fillable contact structures.

Multiple realizations of the 3-sphere as a transverse intersection.

Abstract

By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic structures (eg, they have canonical Stein-fillable contact structures), they also have extrinsic structures of a knot-theoretical nature (eq, the 3-sphere arises in infinitely many distinct ways). This survey is not comprehensive; a number of questions are left open for future work.