Quantum invariants of knotoids

TL;DR

This paper develops quantum invariants for Morse knotoids, extending classical polynomial invariants through quantum state sum models and solutions to the Yang-Baxter equation.

Contribution

It introduces a framework for defining quantum invariants of Morse knotoids, including new polynomial invariants derived from quantum state sums.

Findings

Recovered classical invariants like the bracket polynomial.

Defined new invariants such as the rotational and binary bracket polynomials.

Established connections to the Homflypt polynomial and Alexander polynomial.

Abstract

In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.