6. Unknot or knot to be?

Abstract

‘Unknot or knot to be?’ explains that a knot is a smooth, simple, closed curve in 3D space. Being simple and closed means the curve does not cross itself except that its end returns to its start. All knots are topologically the same as a circle; what makes a circle knotted—or not—is how that circle has been placed into 3D space. The central problem of knot theory is a classification theorem: when is there an ambient isotopy between two knots or how do we show that no such isotopy exists? Key elements of knot theory are discussed, including the three Reidemeister moves, prime knots, adding knots, and the Alexander and Jones polynomials.