zbMATH — the first resource for mathematics

Complex orientations of \(M\)-curves of degree 7. (English) Zbl 0993.14021
Turaev, V. (ed.) et al., Topology, ergodic theory, real algebraic geometry. Rokhlin’s memorial. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 202(50), 215-227 (2001).
The main goal of the paper under review is to prove the non-realizability of two complex \(M\)-schemes of plane real algebraic curves of degree 7. Namely, with the usual notations, it is shown that there is no curve of degree 7 with complex scheme \([J\cup 2_+\cup 1_-\cup 1_-[6_+ \cup 5_-]]\) or \([J\cup 8_+\cup 4_-\cup 1_-[1_+\cup 1_-]]\). The methods of this paper, essentially the Murasugi-Tristram inequality applied to a suitable link, allow also to exclude the realizability of two more schemes: \([J\cup 6_+\cup 2_+\cup 1_-[3_+\cup 3_-]]\) and \([J\cup 5_+\cup 1_+\cup 1_-[4+ \cup 4_-]]\), but do not suffice to exclude \([J\cup 7_+\cup 3_+\cup 1_-[2_+\cup 2]]\). The author announced the exclusion of this scheme in a foregoing paper by using new techniques based on unitary representations of braid groups. This will complete the classification of complex \(M\)-schemes of degree 7.
For the entire collection see [Zbl 0961.00011].

14P25 Topology of real algebraic varieties
57M25 Knots and links in the \(3\)-sphere (MSC2010)