Another way to define knot equivalence is that two knots are equivalent when there is a continuous family of homeomorphisms {ht: R3 → R3 for 0 ≤ t ≤ 1 of space onto itself, such that the last one of them carries the first knot onto the scond knot.
The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots.