Why is ambient isotopy the preferred invariant of knots as opposed to isotopy?
From wikipedia:
Two mathematical knots are equivalent if one can be transformed into the other via a deformation of $\mathbb{R}^3$ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
Why not just have the two knots related to eachother by an isotopy?? Why get the ambient space involved like this? Thanks!