Quantum invariant

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][3]

List of invariants

• Finite type invariant
• Kontsevich invariant
• Kashaev's invariant
• Witten–Reshetikhin–Turaev invariant (Chern–Simons)
• Invariant differential operator[4]
• Rozansky–Witten invariant
• Vassiliev knot invariant
• Dehn invariant
• LMO invariant[5]
• Turaev–Viro invariant
• Dijkgraaf–Witten invariant[6]
• Reshetikhin–Turaev invariant
• Tau-invariant
• I-Invariant
• Klein J-invariant
• Quantum isotopy invariant[7]
• 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[1]
• 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

References

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

Further reading

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

External links

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