Finite type invariants for knotoids

TL;DR

This paper extends finite type invariants from classical knots to knotoids using two approaches, revealing non-trivial invariants for spherical knotoids and classifying type-1 invariants.

Contribution

It introduces two methods to define finite type invariants for knotoids and provides a complete classification of type-1 invariants for spherical knotoids.

Findings

Type-1 invariants are non-trivial for spherical knotoids.

A complete classification of linear chord diagrams of order one.

Examples from affine index polynomial and extended bracket polynomial.

Abstract

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.