# Jones polynomial and knot transitions in topological semimetals

## TL;DR

This paper introduces the use of the Jones polynomial as a topological invariant to analyze and classify knot structures in topological semimetals, revealing how nodal line evolutions relate to changes in knot topology.

## Contribution

It develops a novel framework applying the Jones polynomial to topological semimetals, linking local nodal line interactions to global knot topology changes.

## Key findings

- Jones polynomial captures global knot topology of nodal lines
- Nodal chain semimetals can evolve into Hopf-link configurations
- Theory extends to non-Hermitian exceptional line semimetals

## Abstract

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this paper, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf-link. We extend our theory to 3D non-Hermitian multi-band exceptional line semimetals.