Computations of the slice genus of virtual knots

A virtual knot is an equivalence class of embeddings of $S^1$ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of $S^1$ into a thickened surface $\Sigma_g \times I$; what is the minimal genus of oriented surfaces $S \hookrightarrow M \times I$ with the embedded $S^1$ as boundary, where $M$ is an oriented $3$-manifold with $\partial M = \Sigma_g$? We compute and estimate the slice genus of all virtual knots of $4$ classical crossings or less. We also compute or estimate the slice genus of $46$ virtual knots of $5$ and $6$ classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum.