Knot theory
Geometry and topology
The study of closed loops embedded in three-dimensional space, considered equivalent when one can be deformed into the other without passing the loop through itself.
What is usually left out
The loops must be closed. An open piece of string can always be untangled, so the mathematics only becomes interesting once the ends are joined. The central difficulty is proving two knots are different, which requires invariants — quantities preserved by deformation — since failing to find a deformation proves nothing.