Smooth-Rolling Knots

Morton's knots are a family of space curves to which there is no plane simultaneously tangent in three distinct points. This property enables a physical instance to roll on a plane while having at most two contact points with it. These knots are not smooth-rolling, i.e. they require a force to provoke a rolling motion, during which their center of mass oscillates up and down. By drawing a connection between Morton's knots and smooth-rolling Two-Disk Rollers, we design new smooth-rolling knots. These curves preserve the mesmerizing rolling behavior of Morton's knots while requiring only an infinitesimal force to be set in motion. We apply our method to a variety of knots beyond Morton's family, obtaining tightly-winding, smooth-rolling knots with different topologies.