Topology of maximally writhed real algebraic knots

TL;DR

This paper studies real algebraic knots with maximal Viro invariant in projective 3-space, showing they are all topologically equivalent and identifying their knot type.

Contribution

It demonstrates that all maximally writhed real algebraic knots of a given degree are topologically isotopic and explicitly determines their knot type.

Findings

All such knots are topologically isotopic.

Explicit identification of the knot type for maximally writhed knots.

Maximal Viro invariant characterizes the topological class.

Abstract

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in $RP^3$ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree $d$ with the maximal possible value of this invariant. We show that for a given $d$ all such knots are topologically isotopic and explicitly identify their knot type.