we will talk about kronheimer and mrowka's knot concordance invariant, $s^\sharp$. we compute the invariant for various knots. our computations reveal some unexpected phenomena, including that $s^\sharp$ differs from rasmussen's invariant $s$, and that it is not additive under connected sums. we also generalize the definition of $s^\sharp$ to links by giving a new characterization of the invariant in terms of immersed cobordisms.