Real algebraic knots of low degree

TL;DR

This paper investigates rational real algebraic knots in three-dimensional projective space, establishing classification criteria for knots of degree up to five and providing constructions for knots with specified properties.

Contribution

It proves that knots of degree ≤5 are classified by degree and encomplexed writhe, and offers explicit methods to construct rational knots with desired encomplexed writhe.

Findings

Knots of degree ≤5 are classified by degree and encomplexed writhe.

Any smooth knot with ≤4 crossings has a degree ≤6 rational parametrization.

Explicit constructions of rational knots with arbitrary encomplexed writhe are provided.

Abstract

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree $\leq 6$. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.