Some care required here, actually. The topological space that is the circle is topologically identical with the single closed curve that forms the trefoil knot in 3D Euclidean space, even though neither can be transformed into the other. It's not really enough to talk about being able just to distort one into the other without cutting or glueing. In fact you are allowed to cut, provided you subsequently rejoin things. This is one reason why in knot theory we study the space with the knot removed. That's different in these two cases. This is also related to why the 3D version of the Jordan Curve Theorem is in fact not true.
Topology is the area of mathematical study dealing with the properties of objects that are preserved through deformation, twisting, and stretching without tearing or glueing. The objects studied are called Topological Spaces.

For example, if you're not allowed to tear or glue, but just allowed to deform, then a coffee cup (which has a single hole) can be distorted and moulded into a ring doughnut, but can never be distorted and moulded into a ball, or a sugar bowl (which has two handles). There's no way to get rid of the hole, and there's no way to get a second hole.

Topology comes in two flavours: "Point Set Topology" and "Algebraic Topology"

Related topics are Braid Theory and Knot Theory.