Minimal Degree Parameterizations of the Trefoil and Figure-Eight Knots

TL;DR

This paper identifies the minimal degree sequences for the trefoil and figure-eight knots by constructing explicit compact rational parameterizations, advancing understanding of their minimal polynomial representations.

Contribution

It provides explicit minimal degree sequences for these knots by adapting non-compact parameterizations into compact forms, a novel approach in knot theory.

Findings

Explicit minimal degree sequences for trefoil and figure-eight knots

Method to convert non-compact to compact rational parameterizations

Enhanced understanding of polynomial representations of knots

Abstract

This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots to make it compact. This research was conducted at Ramapo College of New Jersey during the summer of 2011 with Dr. Donovan McFeron.