Polygonal approximation of unknots by quadrisecants

Abstract

If a knot $K$ in $\mathbb R^3$ has only finitely many quadrisecants which meet $K$ at finitely many points, we use these points to form a polygonal approximation called the quadrisecant approximation which is denoted by $\widehat K$. The quadrisecant approximation conjecture states that $\widehat K$ has the knot type of $K$ if $K$ is nontrivial. We give examples of unknots, some polygonal and some smooth, having finitely many but at least two quadrisecants, for which the conjecture holds.