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Khovanov homology and exotic $4$-manifolds (2402.10452v2)
Published 16 Feb 2024 in math.GT and math.QA
Abstract: We show that the Khovanov-Rozansky $\mathfrak{gl}2$ skein lasagna module distinguishes the exotic pair of knot traces $X{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact orientable $4$-manifolds. We also present a family of exotic knot traces that seem not directly recoverable from gauge/Floer-theoretic methods. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen $s$-invariant, which are of independent interest. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna $s$-invariant, and a construction of induced maps on Khovanov homology for cobordisms in $k\mathbb{CP}2$.
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