Real Algebraic Knots of Low Degree

Abstract

In this paper we study rational real algebraic knots in RP 3 . We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤ 6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. Date

Maximally writhed real algebraic links

“…Let K be a real algebraic knot of genus 0 and of degree d ≥ 3. Then the following conditions are equivalent: 1 (i) |w(K)| = N d ; (ii) any real plane tangent to K has only real intersections with CK; (iii a ) any generic real plane cuts K at d or d − 2 real points;…”

“…Let X = P 1 × P 1 and let C be a nonsingular algebraic curve on X of bidegree (2, g + 1), g ≥ 0. Let D be a divisor on C which is cut by a generic curve of bidegree (1,1). Then h 1 (D) = 0.…”

Constructing real rational knots by gluing

“…The proof of 1 and 2 is in [2]. For 3, we will reformulate gluing in a way that is more suitable for it and can also be an alternative proof for 1 and 2.…”

Topology of Maximally Writhed Real Algebraic Knots

“…Two real algebraic links are called rigidly isotopic if they belong to the same connected component of the space of smooth real curves of the same degree. A rigid isotopy classification of real algebraic rational curves in P 3 is obtained in [1] up to degree 5 and in [2] up to degree 6. Also we gave in [2] a rigid isotopy classification for genus one knots and links up to degree 6 (here we speak of the genus of the complex curve L rather than the minimal genus of a Seifert surface of RL).…”