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A minimality property for knots without Khovanov 2-torsion

Onkar Singh Gujral, Joshua Wang

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.

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 has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all knots obtainable from by a proper rational tangle replacement. It follows, for example, that unknotting number 1 knots have 2-torsion in their Khovanov homology.
Paper Structure (3 theorems, 6 equations)

This paper contains 3 theorems, 6 equations.

Key Result

Theorem 1

Suppose $K$ is a knot such that there is no $2$-torsion in $\mathop{\mathrm{Kh}}\nolimits(K)$. If $J$ is a knot that differs from $K$ by a proper rational tangle replacement, then

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 2
  • proof : Proof of Corollary \ref{['cor:unknottingNumber1']}
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['thm:mainthm']}