Quantum invariant
{{Short description|Concept in mathematical knot theory}}
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.
{{Cite journal |first=Maxim |last=Kontsevich |title=Vassiliev's knot invariants |journal=Adv. Soviet Math. |volume=16 |year=1993 |page=137}}
{{Cite journal|last=Watanabe|first=Tadayuki|title=Knotted trivalent graphs and construction of the LMO invariant from triangulations|journal=Osaka J. Math.|year=2007|volume=44|issue=2|page=351|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ojm/1183667985|access-date=4 December 2012}}
List of invariants
- Finite type invariant
- Kontsevich invariant
- Kashaev's invariant
- Witten–Reshetikhin–Turaev invariant (Chern–Simons)
- Invariant differential operator{{Cite arXiv |eprint = math/0406194|last1 = Letzter|first1 = Gail|author1-link= Gail Letzter |title = Invariant differential operators for quantum symmetric spaces, II|year = 2004}}
- Rozansky–Witten invariant
- Vassiliev knot invariant
- Dehn invariant
- LMO invariant{{Cite arXiv |eprint = math/0009222|last1 = Sawon|first1 = Justin|title = Topological quantum field theory and hyperkähler geometry|year = 2000}}
- Turaev–Viro invariant
- Dijkgraaf–Witten invariant{{cite web|url=http://hal.archives-ouvertes.fr/docs/00/09/02/99/PDF/equality_arxiv_1.pdf |title=The invariant of Turaev-Viro from Group category |first=Jerome|last= Petit|publisher=hal.archives-ouvertes.fr |date=1999 |access-date=2019-11-04}}
- Reshetikhin–Turaev invariant
- Tau-invariant
- I-Invariant
- Klein J-invariant
- Quantum isotopy invariant{{cite web |url=http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf |title=Generators of -Character Varieties of Arbitrary Rank Free Groups |work=The 7th KAIST Geometric Topology Fair |first=Sean|last=Lawton|date=June 28, 2007 |access-date=13 January 2022 |archive-url=https://web.archive.org/web/20070720181358/http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf |archive-date=20 July 2007 |url-status=dead}}
- Ermakov–Lewis invariant
- Hermitian invariant
- Goussarov–Habiro theory of finite-type invariant
- Linear quantum invariant (orthogonal function invariant)
- Murakami–Ohtsuki TQFT
- Generalized Casson invariant
- Casson-Walker invariant
- Khovanov–Rozansky invariant
- HOMFLY polynomial
- K-theory invariants
- Atiyah–Patodi–Singer eta invariant
- Link invariant{{cite journal
| last1 = Reshetikhin | first1 = N.
| last2 = Turaev | first2 = V. G.
| doi = 10.1007/BF01239527
| issue = 3
| journal = Inventiones Mathematicae
| mr = 1091619
| pages = 547–597
| title = Invariants of 3-manifolds via link polynomials and quantum groups
| volume = 103
| year = 1991}}
See also
References
{{Reflist}}
Further reading
- {{Cite book
|publisher = Princeton University Press
|isbn = 978-0691085777
|location = Princeton, N.J
|title = Topology of 4-manifolds
|author = Freedman, Michael H.
|date = 1990
|ol = 2220094M
|url-access = registration
|url = https://archive.org/details/topologyof4manif0000free
}}
- {{Cite book
|publisher = World Scientific Publishing Company
|isbn = 9789810246754
|title = Quantum Invariants
|author = Ohtsuki, Tomotada
|date = December 2001
|ol = 9195378M
}}
External links
- [https://books.google.com/books?id=yQRDNCJ0iOUC Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev]
{{Knottheory-stub}}