A Counterexample to the Bernhard–Jablan Unknotting Conjecture

Abstract

We show that there is a knot satisfying the property that for each minimal crossing number diagram of the knot and each single crossing of the diagram, changing the crossing results in a diagram for a knot whose unknotting number is at least that of the original knot, thus giving a counterexample to the Bernhard–Jablan Conjecture.

Keywords:

• Unknotting number
• minimal crossing diagram
• SnapPy
• Knotscape

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 57M27
• 57M25

Acknowledgments

The authors wish to thank the Holland Computing Center at the University of Nebraska, which provided the computing facilities on which the bulk of this work was carried out. The second author acknowledges support by NSF grant DMS-1313559.