arXiv subjects

Ian A. Sullivan

Publications and source records attributed to Ian A. Sullivan.

3 recordsLinked to original sources

arXiv2026

Handle decompositions and the 1-dimensional inputs skein lasagna module

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over $\mathbb{Q}$, defined recently by Ren, Wedrich, Willis, Zhang, and the second author. For a $4$-manifold built out of $1$- and $2$-handles, the invariant can be computed in terms of a cabled colimit of Rozansky-Willis homologies, modulo a new relation which we call the lasso relation. We then present some explicit calculations for disk bundles over $S^{2}$, as well as a partial vanishing result for $4$-manifolds of the form $Σ_{g}\times D^{2}$, $g\geq 1$.

math.GT↗

arXiv2025

Bar-Natan skein lasagna modules and exotic surfaces in 4-manifolds

We construct and study the skein lasagna module obtained by importing the Bar-Natan Khovanov homology package. For 4-manifolds satisfying a non-vanishing condition, we produce pairs of exotic surfaces (with boundary) by using the behavior of skein lasagna gluing maps associated to connect sums of 4-manifolds. We show that one internal stabilization is generally not enough for these exotic knotted surfaces, generalizing results of Hayden to 4-manifolds that contain homologically diverse surfaces admitting primitive fillings.

math.GT↗

arXiv2024

Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$

We interpret Manolescu-Neithalath's cabled Khovanov homology formula for computing Morrison-Walker-Wedrich's $\mathrm{KhR}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan's cobordism category. Using categorified projectors, we compute the $\mathrm{KhR}_2$ skein lasagna modules of (manifold, boundary link) pairs $(S^2 \times B^2, \tilde β)$, where $\tilde β$ is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky-Willis invariant for nullhomologous links in $S^2 \times S^1$. As an application, we show that the $\mathrm{KhR}_2$ skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu.

math.GT↗