Knot invariants are properties of knots that remain unchanged under various transformations, such as stretching or twisting, but not cutting. They help mathematicians classify and distinguish different knots. Common examples of knot invariants include the knot group, polynomial invariants like the Jones polynomial, and linking numbers. These invariants are crucial in the field of topology, particularly in the study of 3-manifolds. By analyzing knot invariants, researchers can determine whether two knots are equivalent or if they can be transformed into one another without cutting. This makes knot invariants essential tools in understanding the complex nature of knots.