The Lifting Problem is NP Complete

TL;DR

This paper proves that determining whether a generic surface in a 3-manifold can be lifted to a knotted surface in 4-space is an NP-complete problem, and also presents an efficient algorithm for testing liftability.

Contribution

It establishes the NP-completeness of the lifting problem for generic surfaces and provides an algorithm to decide liftability efficiently.

Findings

Proves the lifting problem is NP-complete.

Develops an algorithm to test liftability of generic surfaces.

Provides a theoretical framework for understanding knotted surfaces in 4-space.

Abstract

Let $M$ be a 3-manifold. Every knotted (embedded) surface in $M \times \R$ can be moved via an ambient isotopy in such a way that its projection into $M$ is a generic surface. A surface is generic if every point on it is either a regular, double or triple value - the transversal intersection of 1, 2 or 3 embedded surface sheets, or a "branch value" that look like Whitney's umbrella. We elaborate on this in Definition 3.1.1. The double values form arcs, and along each arc two long strips of surface intersect. In a knotted surface, the additional $\R$ coordinate distinguishes between the two strips. One of them must be "higher" than the other. We elaborate on this in Definition 3.1.3. The lifting problem is the problem of determining if a \gls{genericsurface} in $M$ can occur as the $M$-projection of a knotted surface in 4-space in $M \times \R$. The main purpose of this thesis is to study the computational aspects of the lifting problem. We will prove that the problem is NP-complete, and devise an efficient algorithm that determines if a generic surface is liftable.