arXiv subjects

Jacob Migdail

Publications and source records attributed to Jacob Migdail.

3 recordsLinked to original sources

arXiv2026

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

We prove that the reduced odd Khovanov homology of a link $L$ is naturally a module over the exterior algebra of the first homology of the link's branched double-cover. We then describe this module structure more geometrically and related it to the odd Khovanov maps induced by link cobordisms. As an application, we will give a combinatorial proof of a recent result of Spyropoulos-Vidyarthi-Zhang about the odd invariant for $2$-knots in the special case where the $2$-knot is a ribbon $2$-knot. Additionally, we will show that Levine-Zemke's main result from their 2019 paper on Khovanov homology and ribbon concordance remains true for odd Khovanov homology with rational coefficients and with coefficients in $\mathbb{Z}_{2^k}$.

math.GT↗

arXiv2025

The Functoriality of Odd Khovanov Homology up to Sign and Applications

In this dissertation, we extend the odd Khovanov bracket to link cobordisms and prove that our construction is functorial up to sign. We then build an odd Khovanov theory for dotted link cobordisms. Out of the dotted theory, a module structure on the odd Khovanov homology of a diagram over the exterior algebra of the diagram's coloring group arises. We finish by using our functoriality result to prove that if $n$ is even or if the knot has even framing, then the odd Khovanov homology of the $n$-cable of a knot admits an action of the Hecke algebra $\mathcal{H}(q^2,n)$ at $q=i$.

math.GT↗

arXiv2024

Functoriality of Odd and Generalized Khovanov Homology in $\mathbb{R}^3\times I$

We extend the generalized Khovanov bracket to smooth link cobordisms in $\mathbb{R}^3\times I$ and prove that the resulting theory is functorial up to global invertible scalars. The generalized Khovanov bracket can be specialized to both even and odd Khovanov homology. Particularly by setting $\pi=-1$, we obtain that odd Khovanov homology is functorial up to sign. We end by showing that odd Khovanov homology is not functorial under smooth link cobordisms in $S^3\times I$.

math.GT↗