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Torsion of the Khovanov homology
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abstract
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of the Khovanov homology. Based on our calculations, we formulate several conjectures about the torsion and prove weaker versions of the first two of them. In particular, we prove that all non-split alternating links have their integer Khovanov homology almost determined by the Jones polynomial and signature. The only remaining indeterminacy is that one cannot distinguish between $\mathbb{Z}_{2^k}$ factors in the canonical decomposition of the Khovanov homology groups for different values of $k$.
Forward citations
Cited by 2 Pith papers
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Remarks on some infinitesimal symmetries of Khovanov--Rozansky homologies in finite characteristic
A p-differential algebra argument reproves base point independence for characteristic-p Khovanov-Rozansky homology and yields new sl2-symmetry consequences for link homology.
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Traceless $\mathrm{SU}(2)$ characters and $\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots
For two-bridge knots every traceless SU(2) character is binary-dihedral; for (3,n)-torus knots the characters are mostly non-dihedral, with gradings that predict when knot-instanton homology shrinks below the chain complex.
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