Two knots are isotopic if one can be continuously deformed into the other without cutting. The shape changes. The structure persists.
This is a trefoil knot under ambient isotopy. Drag the deformation slider to reshape it — stretch it, compress it, twist its presentation. The knot passes through five distinct configurations, each looking different, each topologically identical.
The invariants on the left cannot change under continuous deformation. No matter how you reshape the knot, its crossing number stays 3, its bridge number stays 2, and it remains a trefoil. These quantities are what make the knot the knot. Everything else is embedding — how the same structure sits differently in space.
That's what isotopy means. The structure is preserved through the transformation.