A minimality property for knots without Khovanov 2-torsion

TL;DR

This paper investigates the relationship between the absence of 2-torsion in Khovanov homology and minimality properties, providing evidence supporting Shumakovitch's conjecture and implications for unknotting number 1 knots.

Contribution

It establishes a minimality property for knots without 2-torsion in their Khovanov homology, linking torsion absence to homological rank minimality under rational tangle replacements.

Findings

Knots without 2-torsion have minimal reduced Khovanov homology rank among related knots.

Unknotting number 1 knots necessarily have 2-torsion in their Khovanov homology.

Supports Shumakovitch's conjecture that all nontrivial knots have 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.