The Unknotting Problem and Normal Surface Q-Theory

TL;DR

This paper extends Q-theory for normal surfaces in 3-manifolds, proving the existence of essential discs at vertices of the Q-projective solution space in boundary reducible cases like unknot complements.

Contribution

It demonstrates that boundary reducible 3-manifolds, such as unknot complements, have essential discs at vertices of the Q-projective solution space, filling a gap in previous theorems.

Findings

Essential discs exist at vertices of the Q-projective solution space for boundary reducible 3-manifolds.

The result applies specifically to unknot complements.

Provides a new understanding of normal surface theory in boundary reducible cases.

Abstract

Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose $M$ is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if $M$ contains an essential surface, then the projective solution space has an essential surface at a vertex. One interesting situation not covered by this theorem is when $M$ is boundary reducible, e.g. $M$ is an unknot complement. We prove that in this case $M$ has an essential disc at a vertex of the Q-projective solution space.