New Quantum Invariants of Planar Knotoids

TL;DR

This paper introduces new polynomial-time computable invariants for planar knotoids, enhancing the classification of open curve knots and providing measures of knottedness in open curves.

Contribution

It develops biframed planar knotoids and constructs novel invariants, improving classification and understanding of open curve knots.

Findings

Improved classification of planar knotoids with up to five crossings

Distinguished several pairs of prime knotoids previously conjectured to be equivalent

Provided polynomial-time computable invariants for biframed planar knotoids

Abstract

In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is knotted. We define biframed planar knotoids, and construct new invariants of these objects that can be computed in polynomial time. As an application of these invariants we improve the classification of planar knotoids with up to five crossings by distinguishing several pairs of prime knotoids that were conjectured to be distinct by Goundaroulis et al.