The topological invariant that describes how many times strands cross in circular DNA is the linking number.
For a covalently closed circular DNA molecule, the linking number (Lk) records how the two DNA strands are linked. It is a topological quantity, so it cannot change unless the DNA backbone is broken. This makes it especially useful for describing plasmids and other closed DNA molecules.
Linking number is related to twist and writhe by the Călugăreanu–White–Fuller relationship: Lk = Tw + Wr. Twist describes how the two strands wind around each other locally, while writhe describes the coiling of the DNA’s axis in space. Supercoiling is a physical state that often results when linking number differs from its relaxed value; it is not the invariant itself.
A common mix-up is treating linking number as a simple visual crossing count. Crossings are counted algebraically, with their orientation, so positive and negative crossings can cancel.