Quadrisecants and essential secants of knots: with applications to the geometry of knots

TL;DR

This survey explores quadrisecants and essential secants in knots, illustrating their historical development and applications to understanding knot geometry through measures like curvature, ropelength, and distortion.

Contribution

It provides a comprehensive overview of how quadrisecants and essential secants relate to knot geometry, highlighting their applications in measuring curvature, ropelength, and distortion.

Findings

Quadrisecants provide bounds on knot complexity.

Essential secants relate to the knottedness of curves.

Applications include estimates of total curvature and ropelength.

Abstract

A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot.