Mechanical normal forms of knots and flat knots

摘要

We describe a series of physical experiments with knotted resilient wires and observe that ambient isotopy classes of knots with a small number of crossing have unique normal forms (corresponding to minima of the energy of the wire). We then describe local moves that the wire knots can perform (Reidemeister moves, Markov moves, the Whitney trick). Further, we consider flat knots (i.e. wire knots squeezed between parallel planes) and prove theorems about their mathematical models.