---
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|AP2ZDXANKC]] 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.