Pointed quandles of linkoids

TL;DR

This paper introduces fundamental pointed quandles for linkoids, enhancing linkoid invariants and distinguishing linkoids with identical under-closures, while also classifying certain homogeneous quandles.

Contribution

It defines fundamental pointed quandles for linkoids and explores their properties, providing new invariants and classifying finite homogeneous quandles.

Findings

Fundamental pointed quandles encode linkoid information.

2-pointed quandles distinguish certain linkoids with identical under-closures.

Classification of all finite $ extinfty$-homogeneous quandles.

Abstract

In this paper we define the fundamental quandle of knotoids and linkoids and prove that it is invariant under the under forbidden-move and hence encodes only the information of the underclosure of the knotoid. We then introduce $n$-pointed quandles, which generalize quandles by specifying $n$ elements as ordered basepoints. This leads to the notion of fundamental pointed quandles of linkoids, which enhances the fundamental quandle. Using $2$-pointed quandle allows us to distinguish 1-linkoids with equivalent under-closures and leads to a couple of 1-linkoid invariants. We also generalize the notion of homogeneity of quandles to $n$-homogeneity of quandles. We classify all $\infty$-homogeneous, finite quandles.