Counterexamples to the quadrisecant approximation conjecture

TL;DR

This paper disproves a conjecture by showing that quadrisecant approximations of knots can either have self-intersections or different knot types, revealing limitations in the previous understanding of quadrisecant approximations.

Contribution

The authors provide counterexamples demonstrating that quadrisecant approximations do not always preserve knot type or remain simple knots, challenging prior assumptions.

Findings

Quadrisecant approximation can have self-intersections.

Quadrisecant approximation can change the knot type.

Counterexamples exist in every knot type.

Abstract

A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self intersections while the quadrisecant approximation of the other knot is a knot with different knot type.