Computation of the knot Floer complex of knots of thickness one

We develop and implement an algorithm that computes the full knot Floer complex of knots of thickness one. As an application, by extending this algorithm to certain knots of thickness two, we show that all but finitely many non-integral Dehn surgery slopes are characterizing for most knots with up to 17 crossings.

💡 Research Summary

The paper introduces a concrete algorithm for computing the full knot Floer complex (CFK) of knots whose thickness is at most one, and it implements this algorithm in SageMath. Knot Floer homology, originally defined by Rasmussen and independently by Ozsváth–Szabó, is a powerful invariant that captures subtle topological information such as fiberedness, genus, and concordance. While many programs can compute the homology groups, none previously output the entire filtered chain complex for an arbitrary knot in S³.

The authors begin by recalling two standard algebraic presentations of CFK: the classical F