Quadrisecant approximation of hexagonal trefoil knot

TL;DR

This paper introduces the quadrisecant approximation method for hexagonal trefoil knots, showing that it preserves the knot type and involves only three quadrisecants, providing insights into knot simplification.

Contribution

It demonstrates that for hexagonal trefoil knots, the quadrisecant approximation preserves the knot type using only three quadrisecants, a novel result in knot theory.

Findings

Hexagonal trefoil knots have exactly three quadrisecants.

Quadrisecant approximation preserves the knot type of the original.

The method simplifies the knot while maintaining its topological properties.

Abstract

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.