# Circuit Complexity of Knot States in Chern-Simons theory

## TL;DR

This paper calculates an upper bound on the circuit complexity of knot states in 3D Chern-Simons theory, revealing connections to number theory, topology, and path integral optimization.

## Contribution

It introduces a method to bound the circuit complexity of knot states using modular transformations and relates it to continued fractions and topological features.

## Key findings

- Upper bound on knot state complexity saturates semiclassically
- Complexity minimization leads to regular continued fractions
- Connections established between complexity, Farey tessellation, and topological features

## Abstract

We compute an upper bound on the circuit complexity of quantum states in $3d$ Chern-Simons theory corresponding to certain classes of knots. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the $3$-sphere of arbitrary torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the $S$ and $T$ modular transformations of the torus as fundamental gates. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. The results are then generalized for a family of multi-component links that are obtained by "Hopf-linking" different torus knots. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies.