Homological Casson type invariant of knotoids

TL;DR

This paper introduces a homological invariant for knotoids, extending classical knot invariants, and demonstrates its effectiveness by establishing bounds on crossing numbers and conditions for knotoid propriety, supported by computed examples.

Contribution

It develops a homology-based invariant for knotoids, providing new tools for their classification and analysis, including bounds and criteria for knotoid properties.

Findings

Provides a lower bound for knotoid crossing number.

Establishes a criterion for proper knotoids in S^2.

Computes invariant values for all prime proper knotoids with up to 5 crossings.

Abstract

We consider an analogue of 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. Value of the extension is a formal sum of subgroups of the first homology group $H_1(\Sigma)$ where $\Sigma$ 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.