Convex Embeddability and Knot Theory
Martina Iannella, Alberto Marcone, Luca Motto Ros, Vadim Weinstein
TL;DR
This work studies convex embeddability for countable linear orders and its extension to circular orders, revealing that the natural quasi-order $L \trianglelefteq L'$ is not a well quasi-order, with continuum-sized chains and antichains and sharp values for unbounding and dominating numbers. It establishes precise complexity bounds for the induced equivalence relations via Borel and Baire reductions, linking convex biembeddability to standard isomorphism while showing several nontrivial complexity gaps (e.g., $E_1$ not reducible to certain LO relations). The authors then transport these order-theoretic insights to knot theory through proper arcs and subarcs, deriving anti-classification results and demonstrating that some knot-relations are not induced by Borel actions, with analogous but stronger phenomena for circularized knots and piecewise subknots. The work also introduces piecewise convex embeddability for circular orders, showing it is strictly more complex than the linear-order case and yielding connections to knot substructure and circular knot operations. Collectively, the results illuminate deep connections between order-theoretic convexity, descriptive set-theoretic complexity, and geometric knot theory, while outlining several open problems and directions for refining these reductions and classifications.
Abstract
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects.