Quantum invariant explained

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.^{[1] [2]}

List of invariants

• Finite type invariant
• Kontsevich invariant
• Kashaev's invariant
• Witten–Reshetikhin–Turaev invariant (Chern–Simons)
• Invariant differential operator^[3]
• Rozansky–Witten invariant
• Vassiliev knot invariant
• Dehn invariant
• LMO invariant^[4]
• Turaev–Viro invariant
• Dijkgraaf–Witten invariant^[5]
• Reshetikhin–Turaev invariant
• Tau-invariant
• I-Invariant
• Klein J-invariant
• Quantum isotopy invariant^[6]
• 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^[7]
• Casson invariant
• Seiberg–Witten invariants
• Gromov–Witten invariant
• Arf invariant
• Hopf invariant

See also

• Invariant theory
• Framed knot
• Chern–Simons theory
• Algebraic geometry
• Seifert surface
• Geometric invariant theory

Further reading

• Book: Princeton University Press . 978-0691085777 . Princeton, N.J . Topology of 4-manifolds . Freedman, Michael H. . 1990 . 2220094M . registration .
• Book: World Scientific Publishing Company . 9789810246754 . Quantum Invariants . Ohtsuki, Tomotada . December 2001 . 9195378M .

External links

• Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev

Notes and References

1. Maxim . Kontsevich . Vassiliev's knot invariants . Adv. Soviet Math. . 16 . 1993 . 137.
2. Watanabe. Tadayuki. Knotted trivalent graphs and construction of the LMO invariant from triangulations. Osaka J. Math.. 2007. 44. 2. 351. 4 December 2012.
3. math/0406194. Letzter. Gail. Gail Letzter . Invariant differential operators for quantum symmetric spaces, II. 2004.
4. math/0009222. Sawon. Justin. Topological quantum field theory and hyperkähler geometry. 2000.
5. Web site: The invariant of Turaev-Viro from Group category . Jerome. Petit. hal.archives-ouvertes.fr . 1999 . 2019-11-04.
6. Web site: Generators of \operatorname{SL}( 2 , \C) -Character Varieties of Arbitrary Rank Free Groups . The 7th KAIST Geometric Topology Fair . Sean. Lawton. June 28, 2007 . 13 January 2022 . https://web.archive.org/web/20070720181358/http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf . 20 July 2007 . dead.
7. Reshetikhin . N. . Turaev . V. G. . 10.1007/BF01239527 . 3 . Inventiones Mathematicae . 1091619 . 547–597 . Invariants of 3-manifolds via link polynomials and quantum groups . 103 . 1991.