Efficient calculations of $S$-invariants for links

We describe an algorithm that can effectively calculate the $s$-invariant of a link as defined by Beliakova and Wehrli. Our computations show that this cannot be done by merely calculating the $E_{\mathrm{\infty }}$-page of the Bar-Natan–Lee–Turner spectral sequence. Our methods also work for $s$-invariants coming from sl(3)-link homology.