| A knot is a simple closed curve in R3 . A secant line is a straight line which intersects the knot in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect a knot in at least three, four and five distinct places, respectively. It is clear that any closed curve has secants. A little thought will reveal that nontrivial knots must have trisecants, but they do not necessarily have quintisecants. The relationship between knots and quadrisecants is not so immediately clear. In 1933, E. Pannwitz proved that nontrivial generic polygonal knots have at least one quadrisecant. In 1994, G. Kuperberg showed that all (nontrivial tame) knots have at least one quadrisecant.; Quadrisecants come in three basic types. These are distinguished by comparing the orders of the four points along the knot and along the quadrisecant line. These three types are labeled simple, flipped and alternating. It turns out that alternating quadrisecants capture the knottedness of a knot.; The Main Theorem shows that every nontrivial tame knot in R3 has an alternating quadrisecant. This result refines the previous work about quadrisecants and gives greater geometric insight into knots. The Main Theorem provides new proofs to two previously known theorems about the total curvature and second hull of knotted curves. Moreover, essential alternating quadrisecants may be used to dramatically improve the known lower bounds on the ropelength of thick knots. |
