A polynomial upper bound on Reidemeister moves (1302.0180v2)

Abstract: We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of Reidemeister moves required to transform a diagram of the link into a disconnected diagram.

A Polynomial Upper Bound on Reidemeister Moves for the Unknot

The paper authored by Marc Lackenby, titled "A Polynomial Upper Bound on Reidemeister Moves," addresses a fundamental question in knot theory regarding the transformation of knot diagrams using Reidemeister moves. The main result establishes a polynomial upper bound on the number of Reidemeister moves required to transform any diagram of the unknot with $c$ crossings into the trivial diagram and proves a similar bound for split links. Specifically, the paper demonstrates that a diagram of the unknot can be transformed into the trivial diagram using at most $(236c)^{11}$ Reidemeister moves. This finding provides substantial insights into the computational complexity of knot theory problems.

Main Results and Theorems

The core contributions of the paper are encapsulated in two theorems:

1. Theorem 1.1 (Unknot): Given a diagram $D$ of the unknot with $c$ crossings, there exists a sequence of at most $(236c)^{11}$ Reidemeister moves transforming $D$ into the trivial diagram. Each diagram in the sequence contains at most $(7c)^2$ crossings.
2. Theorem 1.2 (Split Links): When $D$ represents a diagram of a split link with $c$ crossings, a sequence of at most $(49c)^{11}$ Reidemeister moves will transform $D$ into a disconnected diagram. The maximum number of crossings in each diagram of the sequence is bounded by $9c$.

These results are significant because they establish a polynomial, rather than exponential, bound, thus marking a step towards understanding the complexity of knot recognition problems.

Technical Insights and Methods

The paper heavily leverages previous work by Dynnikov on knot arc presentations. Arc presentations are representations of knots emphasizing grid diagrams and facilitate understanding of transformations using Reidemeister moves. Lackenby builds upon Dynnikov's approach, focusing on minimizing the so-called arc index and utilizing properties of normal surfaces in 3-manifolds:

• Arc Index Decomposition: The paper employs arc presentations to offer a systematic way of breaking down a knot or link diagram into smaller, manageable components, allowing for efficient movement and reduction through Reidemeister moves.
• Normal Surface Theory: Applying normal surface theory, the paper draws on complex transformation techniques facilitated by surfaces such as discs and spheres in polyhedral decompositions. It uses normal surfaces to strategically plan the series of Reidemeister moves, drawing connections between abstract mathematical constructs and practical iteration.
• Branched Surfaces: Overcoming technical challenges, the study introduces branched surfaces as a method to handle the intricacies of surface manipulation within 3-manifolds. This involves detailed constructions of branched surfaces to track and guide the transformation sequence.

Implications and Future Directions

While the polynomial upper bound is a substantial development, the paper notes that the established degree of the polynomial can be subject to improvement. The techniques outlined in the paper extend potential applications beyond the unknot to more complex knot types, although such extensions will require overcoming considerable mathematical challenges.

The work prompts further inquiry into whether a polynomial time algorithm could definitive recognize the unknot, highlighting the broader Turing complexity within knot theory and computational topology. Although this remains speculative and contentious given current complexity assumptions, the framework and methods described may offer a groundwork for future breakthroughs.

Overall, Lackenby's paper significantly impacts both theoretical knot theory and algorithmic applications, providing a new angle on the pursuit to effectively resolve and classify knots using elementary moves captured in topological spaces.