Counting Generic Quadrisecants of Polygonal Knots

TL;DR

This paper establishes an upper bound on the number of generic quadrisecants intersecting a polygonal knot in general position, based on the number of edges, advancing understanding of knot geometric properties.

Contribution

It provides a new upper bound for the number of generic quadrisecants of polygonal knots in general position, relating it to the number of edges.

Findings

Derived an explicit upper bound formula.

Applied the bound to various polygonal knot configurations.

Enhanced understanding of geometric constraints in knot theory.

Abstract

Let $K$ be a polygonal knot in general position with vertex set $V$. A \emph{generic quadrisecant} of $K$ is a line that is disjoint from the set $V$ and intersects $K$ in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot $K$ in general position. This upper bound is in terms of the number of edges of $K$.