On the Concordance Invariants of Cable Links

We study the Beliakova–Wehrli link $s$-invariant and Cavallo’s link $\tau$-invariant of coherently oriented cables. For knots satisfying extremal four-ball genus and maximal self-linking hypotheses, we establish bounds for the $s$-invariant of $(p,q)$-cables, where $p$ and $q$ are not necessarily coprime, and show that these bounds are sharp under certain conditions. For the $\tau$-invariant, we prove analogous bounds for arbitrary companion knots and exact formulas whenever $\tau (K)=\pm g_4(K)\mathrm{\ne }0$. Specializing to the coherently oriented cable links associated to an $n$-framed knot, we apply these bounds to the lasagna $s$-invariant of Ren and Willis. The resulting applications provide upper bounds for the lasagna $s$-invariant of knot traces and a criterion for distinguishing homeomorphic zero traces up to orientation-preserving diffeomorphism.