After 40 years, we proved that if you untangle one knot into another, the physics of it forces the result to be simpler.
March 30, 2026
Original Paper
Simon's knot genus problem and Lewin $3$-manifold groups
arXiv · 2603.26580
The Takeaway
In knot theory, complexity is measured by the 'genus' of the surface it forms. This proof confirms that a specific mathematical connection between two knots acts like a one-way street, ensuring that the resulting knot can never be more tangled than the one it started from, establishing a long-sought law of 'simplicity' in topology.
From the abstract
We provide a positive answer to an old problem of Jonathan K. Simon: if $K$ and $K'$ are two knots such that there is an epimorphism from the knot group of $K$ to the knot group of $K'$, then the genus of $K$ is greater than or equal to the genus of $K'$. We achieve this by proving a conjecture of Friedl and Lück, which states that the existence of a map between admissible $3$-manifolds that induces an epimorphism on the fundamental groups and an isomorphism on the rational homologies yields an