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A Kirby color for Khovanov homology

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arxiv 2210.05640 v2 pith:XXRJYBJZ submitted 2022-10-11 math.GT math.QAmath.RT

Matthew Hogancamp , David E. V. Rose , Paul Wedrich This is my paper · ORCID
classification math.GTmath.QAmath.RT
keywords homologykhovanovhandlekirbycolorinvariantisomorphismkirby-colored
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abstract

We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu--Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the $\mathfrak{gl}_{2}$ skein lasagna module, hence is an invariant of $4$-dimensional $2$-handlebodies.

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  1. From Link Homology to Topological Quantum Field Theories

    math.QA 2025-09 accept novelty 2.0 of 10

    A survey of how link homology theories extend to 4-manifold invariants called skein lasagna modules, which can distinguish exotic smooth structures.

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