One isotopy invariant is tricoloration, or tricolorability. It deals with the ability to use three different "colors" to color a knot. A knot is colored by individual strands, where a strand is the part of a two-dimensional representation of a knot between undercrossings. For instance, each strand of the knot below is colored differently.
A knot is tricolorable if:
Rule 1: At every crossing, either all three strands are of a different color, or the same color.
Rule 2: All three colors are used in coloring the knot.
It is fairly simple to prove that tricolorability is an isotopy invariant. Below are images of every kind of Reidemeister move with tricoloration maintained. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant.