The Ribbon Bound

Ribbon concordance is a partial order on knots: K_1 ≤ K_2 if there’s a concordance (cobordism in S^3 × I) with only saddle points oriented one way — ribbon singularities, no caps. Gordon asked whether this partial order has infinite descending chains.

For fibered knots, it doesn’t (arXiv:2603.10884). Simplicial volume and dilatation (a measure of the complexity of the monodromy homeomorphism) are both monotone under ribbon concordance. Since dilatation is a positive real number that can only decrease along a ribbon concordance chain, and the set of dilatations of fibered knots is well-ordered, every fibered knot has only finitely many ribbon-concordance predecessors.

The structural insight: the constraint is not topological but dynamical. Fibered knots carry surface homeomorphisms (the monodromy), and the dilatation measures the dynamical complexity of this homeomorphism. Ribbon concordance can only simplify the dynamics — it cannot increase dilatation. The finiteness of predecessors follows not from knot theory but from the well-ordering of a dynamical invariant. The topology constrains the dynamics, and the dynamics constrains the order.