Quantum invariant

In the mathematical field of knot theory , quantum invariants are invariants of knots, links and 3-manifolds that are defined by means of representation theory of quantum groups or, more generally, from solutions of the Yang-Baxter equation .

Construction via R-matrices

An R matrix and an isomorphism as well as a Markov trace are given . ${\ displaystyle R = (R_ {kl} ^ {ij}) _ {i, j, k, l} \ colon V \ otimes V \ to V \ otimes V}$ ${\ displaystyle \ mu \ colon V \ to V}$ ${\ displaystyle Tr}$

By means of the R-matrix defines one representations of the braid groups by ${\ displaystyle B_ {n}}$

{\ displaystyle \ rho _ {n} \ colon B_ {n} \ to Aut (V ^ {\ otimes n})}

{\ displaystyle \ rho _ {n} (\ sigma _ {i}) = id_ {V} ^ {i-1} \ otimes R \ otimes id_ {V} ^ {ni-1}}

.

Using Alexander's theorem , every loop can be represented as the end of a braid. By means of the Markov trace one obtains an invariant from the endomorphism defined in this way and with Markow's theorem one can show that this invariant is well defined.

Examples of quantum invariants

Kontsevich invariant
Kashaev invariant
Witten – Reshetikhin – Turaev invariant (see also: Chern-Simons functional )
Rozansky – Witten invariant
LMO invariant
Turaev – Viro invariant
Dijkgraaf – Witten invariant
Reshetikhin – Turaev invariant
Casson-Walker invariant
HOMFLY polynomial
Vassiliev invariants (finite-type invariants)

Web links

Garoufalidis: Quantum Knot Invariants

literature

Reshetikhin, N .; Turaev, VG: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991) no. 3: 547-597.
Kirby, Robion; Melvin, Paul: The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl (2, C). Invent. Math. 105 (1991) no. 3, 473-545.
Ohtsuki, Tomotada: Quantum invariants. A study of knots, 3-manifolds, and their sets. Series on Knots and Everything, 29. World Scientific Publishing Co., Inc., River Edge, NJ, 2002. ISBN 981-02-4675-7
Turaev, Vladimir G .: Quantum invariants of knots and 3-manifolds. Second revised edition. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 2010. ISBN 978-3-11-022183-1