A minimality property for knots without Khovanov 2-torsion

Abstract

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology. We show that if a knot $K$ has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all knots obtainable from $K$ by a proper rational tangle replacement. It follows, for example, that unknotting number 1 knots have 2-torsion in their Khovanov homology.

arXiv comment3 pages