Everywhere equivalent and everywhere different knot diagrams

Abstract

A knot diagram is said to be everywhere different (resp. everywhere equivalent) if all the diagrams obtained by switching one crossing represent different (resp. the same) knot(s). We exhibit infinitely many everywhere different knot diagrams. We also present several constructions of everywhere equivalent knot diagrams, and prove that among certain classes these constructions are exhaustive. Finally, we consider a generalization to link diagrams, and discuss some relation to symmetry properties of planar graphs.