---
tag: YGPR4VR3GP
kind: concept
name: Quantum invariant of 3-manifolds
msc: [57K16, 57K31]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_invariant
---

## Summary

Quantum invariants of $3$-manifolds are invariants of closed oriented [[3-manifolds|R20TB2KEQQ]].
They are built from the representation theory of quantum groups at roots of unity, or more generally from modular tensor categories.
The main examples are the Witten–Reshetikhin–Turaev invariants, which Witten predicted from quantum [[Chern–Simons theory|F28G886PWD]] and Reshetikhin and Turaev constructed rigorously.
They are computed from a surgery presentation of the manifold, using colored versions of the [[Jones polynomial|4SPC17FPXE]].
They are part of [[topological quantum field theories|DAP92B80J2]] in dimension $3$.

## Explanations

### L1 human

Every closed oriented $3$-manifold can be obtained by surgery on a link in the $3$-sphere, but in many different ways.
A quantum invariant assigns a number to the link, built from knot polynomials, and corrects it so that the number does not depend on which link is chosen.
The result is a number attached to the manifold itself.
These numbers can be computed from a picture of the link, yet their geometric meaning is still only partly understood.