  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.
