Everywhere equivalent and everywhere different knot diagrams

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.

Keywords

alternating knot, semiadequate knot, Jones polynomial, Kauffman bracket, planar graph, edge transitive

2010 Mathematics Subject Classification

Primary 57M25. Secondary 05C10, 05C75, 57M15.

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Published 21 March 2013