Maximally writhed real algebraic links

Abstract

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots and links in RP 3 which can be viewed as a first order Vassiliev invariant. In this paper we classify real algebraic links of degree d with the maximal value of this invariant in its two versions: w and w λ .

On Osculating Framing of Real Algebraic Links

“…In this paper we add one more item to this collection: we show that the self-linking number of L with respect to the osculating framing attains its maximal value (for links of a given degree) if and only if L is an M W λlink. The proof is very similar to that of the main theorem of [3]. Let us give precise definitions and statements.…”

“…It follows that the equality is attained in all the inequalities used in the proof, in particular, we have | lk(K i , K j )| = lk(K i , K j ) for i = j. Since all components of an M W λ -link endowed with a complex orientation are positively linked (see [3]), we are done. This completes the proof of the "only if " part of (b).…”

“…By Murasugi's result [4,Prop. 7.5] (see also [3,Prop. 1.2]), the number of crossings of any projection of K i is at least (a i + 2)(a i − 1)/2.…”

“…2] (which claims, in particular, that L is an M W λ -link as soon as ps(L) ≥ d − 2). Here we denote with ps(L) the plane section number of L. It is a topological invariant of a link in RP 3 defined in [3] as the minimal number of intersection points with a generic plane where the minimum is taken over the isotopy class of the link.…”

On the Hyperbolicity Locus of a Real Curve

“…, p 3 ) as in Figure 3(right). Such a perturbation is possible due to [1,Theorem 2.4] (see also [3,Lemma 5.1]): one should add the lines one by one. It is easy to see that L, L ′ ∈ H(C).…”