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Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

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arxiv math/0502371 v2 pith:CCEYZFLD submitted 2005-02-17 math.GT math.QA

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keywords khovanov-jacobssontheorylinksnumbersurface-knotbar-natanclassicalinvariant
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abstract

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invariant of a surface-knot which is a generalization of the Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any $T^2$-knot has the trivial Khovanov-Jacobsson number.

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Cited by 1 Pith paper

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  1. A module structure on odd Khovanov homology and the odd invariant for ribbon 2-knots

    math.GT 2026-07 accept novelty 7.0 of 10

    Reduced odd Khovanov homology is a module over Λ*H1(Σ(L)), implying n(F)=|H1(Σ(F))| for ribbon 2-knots and injectivity of ribbon concordances over Q and Z_{2^k}.

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