Generalizations of Knotoids and Spatial Graphs

TL;DR

This paper extends the concept of knotoids to include multiple poles, intervals, and circles, unifying various topological objects and exploring their invariants, while also introducing knotoidal graphs as a generalization of spatial graphs.

Contribution

It introduces generalized knotoids with multiple poles and extends to knotoidal graphs, broadening the scope of topological invariants and their applications.

Findings

Generalized knotoids encompass a wide range of topological objects.

Various invariants like height and polynomial invariants are adapted to the generalized setting.

Introduction of knotoidal graphs as an extension of spatial graphs.

Abstract

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily many poles, intervals, and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalize to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.