## Abstract

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 H_{1}(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 S^{2} 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 S^{2} 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 language | English |
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Article number | 142 |

Journal | Results in Mathematics |

Volume | 76 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2021 |

## Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics

## Keywords

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