Tightening a knot as much as possible creates a shape profile that changes as the knot is allowed to grow.
April 23, 2026
Original Paper
The Ideal Stratum and Deformation Persistence of Knot Types
arXiv · 2604.17905
The Takeaway
Ideal knots represent the tightest possible configuration for a given knot type, like a string pulled into its most compact form. This new framework uses persistence modules to track how the geometry of these knots evolves as the string length increases. Most researchers previously looked for a single perfect shape for each knot. This approach reveals that knots possess a dynamic geometric profile that shifts through different phases. These findings could help biologists understand how long strands of DNA or proteins fold themselves into functional shapes in cramped cellular environments.
From the abstract
We introduce a persistent geometric framework for knot types based on normalized spaces of representatives. For a knot type $K$ and a scale parameter $\Lambda>0$, we consider the space \[ Y_\Lambda(K)= R_{1,\Lambda}(K) \] of representatives of $K$ with thickness at least $1$ and length at most $\Lambda$, modulo orientation-preserving reparametrization and rigid motions. This space may be viewed as a normalized moduli-type space of unparametrized representatives of the knot type $K$, equipped wit