# Imaging nodal knots in momentum space through topolectrical circuits

## TL;DR

This paper demonstrates how to realize and probe momentum space knots using reciprocal topolectrical circuits, enabling visualization and topological analysis of complex knotted structures through impedance measurements.

## Contribution

It introduces a practical method to create and analyze momentum space knots in reciprocal circuits, overcoming previous limitations related to non-reciprocity and long-range hoppings.

## Key findings

- Successfully mapped the topological drumhead states in circuits
- Reconstructed knot invariants like the Alexander polynomial
- Experimentally visualized a Hopf-link in a circuit setup

## Abstract

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. Momentum space knots, in particular, have been elusive due to their requisite finely tuned long-ranged hoppings. Even if constructed, probing their intricate linkages and topological "drumhead" surface states will be challenging due to the high precision needed. In this work, we overcome these practical and technical challenges with RLC circuits, transcending existing theoretical constructions which necessarily break reciprocity, by pairing nodal knots with their mirror image partners in a fully reciprocal setting. Our nodal knot circuits can be characterized with impedance measurements that resolve their drumhead states and image their 3D nodal structure. Doing so allows for reconstruction of the Seifert surface and hence knot topological invariants like the Alexander polynomial. We illustrate our approach with large-scale simulations of various nodal knots and an experiment that maps out the topological drumhead region of a Hopf-link.