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 ^[8]
• Casson invariant
• Seiberg–Witten invariant
• 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. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups". Invent. Math. 103 (1): 547. doi:10.1007/BF01239527. Retrieved 4 December 2012.
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. [math/0406194] Invariant differential operators for quantum symmetric spaces, II
5. [math/0009222v1] Topological quantum field theory and hyperk\"ahler geometry
6. http://hal.archives-ouvertes.fr/docs/00/09/02/99/PDF/equality_arxiv_1.pdf
7. http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf
8. Invariants of 3-manifolds via link polynomials and quantum groups - Springer

Further reading

• Freedman, Michael H. (1990). Topology of 4-manifolds. Princeton, N.J: Princeton University Press. ISBN 0691085773.
• Ohtsuki, Tomotada (December 2001). Quantum Invariants. World Scientific Publishing Company. ISBN 9789810246754.

External links

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