Encomplexing the writhe (Q2735122)

In topology, one defines the writhe of a knot \(K\) with respect to a general linear projection \(p\) of the knot on the plane [see \textit{L. H. Kauffman}, Am. Math. Mon. 95, No. 3, 195-242 (1988; Zbl 0657.57001)]. The writhe is the sum of the local writhes at all double points of the projection \(p(K)\) of \(K\). The local writhe at a double point of \(p(K)\) is equal to \(\pm 1\). Now, unfortunately, the local and global writhes depend on the projection \(p\). Indeed, the local and global writhes are easily seen to be not invariant under the first Reidemeister move on \(p(K)\) (i.e. the move that straightens out a double point of \(p(K)\) of the form of the letter \(\alpha\)). NEWLINENEWLINENEWLINEIn the paper under review, the author defines an enhanced writhe in case the knot \(K\) is real algebraic. The idea is that the Zariski closure of the projection \(p(K)\) can have isolated real points like the real algebraic curve given by the equation \(y^2=x(x+1)^2\). The author defines a local writhe for these isolated real points. He defines the encomplexed writhe of \(K\) as the sum of the local writhes of \(p(K)\), the sum being taken over all double points of \(p(K)\) and all isolated real points of the Zariski closure of \(p(K)\). The author shows that the encomplexed writhe of a real algebraic knot does not depend on the projection. NEWLINENEWLINENEWLINEAt the end of the paper, some generalizations are proposed. The author proposes also to encomplexe the Smale invariant of an immersion of \(S^{2n}\) into \({\mathbb R}^{4n}\). The real algebraic analogue of such an immersion should be, according to the author, a generic real regular map of \(S^{2n}\) into a real Abelian variety \(A\). The enhanced Smale invariant, then, would take into account the isolated real points of the Zariski closure of the image of \(S^{2n}\) in \(A\). However, as far as I can see, the only real regular maps from \(S^{2n}\) into a real Abelian variety are the constant maps. Therefore, the discussion of the enhanced Smale invariant should probably be taken with a grain of salt.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].