Polynomial-time unknot detection

Determine whether there exists a polynomial-time algorithm that decides, given a knot diagram, whether it represents the unknot; equivalently, develop or rule out polynomial-time unknot recognition.

Background

Unknot detection is a central computational problem in knot theory. Although there are algorithms that decide whether a diagram represents the unknot, their known guarantees do not establish polynomial-time complexity. The authors emphasize the significance of this unresolved complexity question and its relation to hard unknot diagrams.

Hard unknot diagrams, which require crossing number increase before simplification, may challenge or inform algorithmic approaches. The paper compiles a large dataset of such diagrams, potentially useful for benchmarking and probing the limits of heuristic and algorithmic simplification strategies.