Quadrisecants and essential secants of knots: a survey

A quadrisecant line is one which intersects a curve in at least four points. In this survey talk, we review the history and current state of knowledge about quadrisecants. We give key ideas behind the proof that non-trivial knots must have an alternating quadrisecant line. This includes the idea of an essential secant, which captures something about the knottedness of a knot. If time allows, we show how these ideas can be applied to questions about the geometry of a knot (for example total curvature and ropelength).