A Homological Casson Type Invariant of Knotoids

Vladimir Tarkaev

Research output: Contribution to journalArticlepeer-review


We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group H1(Σ) where Σ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in S2 into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in S2 to be a proper knotoid (or pure knotoid with respect to Turaev’s terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

Original languageEnglish
Article number142
JournalResults in Mathematics
Issue number3
StatePublished - Aug 2021

Scopus subject areas

  • Mathematics (miscellaneous)
  • Applied Mathematics


  • Casson knot invariant
  • crossing number
  • first homology group
  • invariants of knotoids
  • Knotoid
  • proper knotoid


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