Effective bounds on characterising slopes for all knots

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