Inventiones mathematicae

, Volume 216, Issue 1, pp 125–152 | Cite as

Maximally writhed real algebraic links

  • Grigory MikhalkinEmail author
  • Stepan Orevkov


Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots and links in \(\mathbb {RP}^3\) which is not a topological isotopy invariant. In this paper we study real algebraic links of degree d with the maximal value of this invariant. We show that these links admit entirely topological description. In particular, these links are characterized by the property that any of their planar diagram has at least \((d-1)(d-2)/2-g-1\) crossing points where g is the genus of the complexification. Also we show that these links are characterized by the property that any generic plane intersects them in at least \(d-2\) real points. In addition we give a complete topological classification of these links.



We are grateful to Oleg Viro for very useful and fruitful discussions.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveCarougeSwitzerland
  2. 2.Steklov Mathematical InstituteMoscowRussia
  3. 3.IMT, Université Paul SabatierToulouseFrance

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