Knot Groups

Knot groups are mathematical tools used to study knots—loops of string tangled in space—by examining their fundamental structure. Specifically, a knot group is the set of all possible ways (called "paths") to move around the knot without crossing it, organized so that combining paths reflects following one after another. This group captures essential features of the knot's shape and complexity, serving as an algebraic signature that helps distinguish one knot from another. In essence, knot groups translate the geometric intricacies of tangled loops into a form that mathematicians can analyze and compare systematically.