# Knot Topology in Quantum Spin System

## TL;DR

This paper introduces knot theory as a novel approach to understanding topological phases in quantum spin systems, mapping Majorana modes to knots and links to visualize and characterize their properties.

## Contribution

It applies knot theory to quantum spin systems, providing a new geometric perspective on topological phases and their eigenstate configurations.

## Key findings

- Majorana modes mapped to knots and links
- Topological properties visualized using crossing and linking numbers
- Eigenstate curves form knots in gapless phases

## Abstract

Knot theory provides a powerful tool for the understanding of topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in the quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of the phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.