Unsolvable problems about higher-dimensional knots and related groups

TL;DR

This paper demonstrates that determining whether certain fundamental groups of codimension 2 embeddings belong to specific subclasses is algorithmically impossible, highlighting fundamental limits in classifying higher-dimensional knots.

Contribution

It proves the undecidability of membership problems for classes of fundamental groups associated with higher-dimensional knot complements.

Findings

Membership problems are algorithmically unsolvable for these classes.

The results apply to various types of codimension 2 embeddings.

This establishes fundamental limits in the algorithmic classification of higher-dimensional knots.

Abstract

We consider classes of fundamental groups of complements of various kinds of codimension 2 embeddings and show that, in general, the problem of deciding whether or not a group in one class belongs to a smaller class is algorithmically unsolvable.