A Homological Casson Type Invariant of Knotoids

Autor: Vladimir Tarkaev
Rok vydání: 2021
Předmět:
Zdroj: Results in Mathematics. 76
ISSN: 1420-9012
1422-6383
DOI: 10.1007/s00025-021-01445-y
Popis: 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 $$H_1(\varSigma )$$ where $$\varSigma $$ 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.
Databáze: OpenAIRE
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