Effective bounds on characterising slopes for all knots

A slope $p\mathrm{/}q$ is characterising for a knot $K\subset {\mathbb{S}}^3$ if the orientation-preserving homeomorphism type of the manifold ${\mathbb{S}}_K^3(p\mathrm{/}q)$ obtained by performing Dehn surgery of slope $p\mathrm{/}q$ along $K$ uniquely determines the knot $K$. We combine new applications of results from hyperbolic geometry with previous individual work of the authors to determine, for any given knot $K$, an explicit bound $\mathcal{C}(K)$ such that $\mathrm{\mid }q\mathrm{\mid }>\mathcal{C}(K)$ implies that $p\mathrm{/}q$ is a characterising slope for $K$. Furthermore, we find an optimal such $\mathcal{C}(K)$ for certain satellite knots with winding number zero patterns.