Polynomial Bounds on Reidemeister Moves in Knot Theory

Explore a challenging problem in knot theory through this 50-minute lecture by Marc Lackenby at BIMSA. Delve into the complexities of determining whether a given knot is trivial and learn about the limitations of using Reidemeister moves to simplify knot diagrams. Discover a groundbreaking algorithm that transforms a trivial knot diagram with n crossings into one with no crossings using at most (236n)^11 Reidemeister moves. Understand how this approach provides a new proof that the problem is in NP and offers a conceptually simple method for determining knot triviality. Examine a recent generalization of this result, which introduces a polynomial bound on the number of Reidemeister moves needed to transform between any two diagrams of a given knot type.