##
**Invariants of 3-manifolds via link polynomials and quantum groups.**
*(English)*
Zbl 0725.57007

Using techniques of quantum field theory, Witten defined, on a physical level of rigor, a series of new invariants of 3-manifolds and of links in 3-manifolds. The present paper provides a mathematical background for the construction of these invariants. For closed 3-manifolds, they form a sequence of complex numbers parametrized by complex roots of unity which specialize to the values of the Jones polynomial at the corresponding roots of unity for a link in the 3-sphere. For manifolds with boundary they form a sequence of finite dimensional complex linear operators producing for each root of unity a 3-dimensional topological quantum field theory. The paper uses the algebraic language of Hopf algebras. A central ingredient of the construction of the new invariants are the Kirby-moves relating different surgery presentations by framed links in the 3-manifold. Certain invariants of framed links in the 3-sphere combine in expressions invariant under these Kirby-moves thus giving the new invariants. The basic invariant used is the Jones polynomial which is known to be connected with the quantum enveloping algebra of the Lie algebra \(sl_ 2({\mathbb{C}})\) and its fundamental representation; other irreducible representations of the algebra are used to construct more basic invariants of links in the 3-sphere, which serve then as a groundstock for the construction of the invariants of general 3- manifolds.

A purely topological and combinatorial construction of the invariants, thus avoiding the algebraic machinery of the present paper, can be found in papers by W. B. R. Lickorish [Pac. J. Math. 149, No.2, 337-347 (1991)] and K. H. Ko and L. Smolinsky [ibid., 319-336 (1991)]. Concrete computations or applications of the new invariants have not yet appeared.

A purely topological and combinatorial construction of the invariants, thus avoiding the algebraic machinery of the present paper, can be found in papers by W. B. R. Lickorish [Pac. J. Math. 149, No.2, 337-347 (1991)] and K. H. Ko and L. Smolinsky [ibid., 319-336 (1991)]. Concrete computations or applications of the new invariants have not yet appeared.

Reviewer: B.Zimmermann (Trieste)

### MSC:

57M99 | General low-dimensional topology |

81T99 | Quantum field theory; related classical field theories |

### Keywords:

Jones polynomial; link; 3-sphere; 3-dimensional topological quantum field theory; Kirby-moves; surgery; quantum enveloping algebra
PDF
BibTeX
XML
Cite

\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)

### References:

[1] | [A] Atiyah, M.: On framings of 3-manifolds. (1989) Preprint · Zbl 0716.57011 |

[2] | [A1] Atiyah, M.: Topological quantum field theories. Publ. Math. IHES,68, 175-186 (1989) · Zbl 0692.53053 |

[3] | [Dr] Drinfeld, V. G.: Quantum groups, Proceedings of the International Congress of Mathematicians. Vol. 1, pp. 798-820. Berkeley: Academic Press 1986 |

[4] | [Dr1] Drinfeld, V. G.: On the almost cocommutative Hopf algebras. Algebra and Analis,1, 30-47 (1989), (in Russian) |

[5] | [FRT] Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantization of Lie algebras and Lie groups, Algebra i Analiz i Analiz1, 1 (1989) (in Russian) |

[6] | [FR] Fenn, R., Rourke, C.: On Kirby’s calculus of links. Topology18, 1-15 (1979) · Zbl 0413.57006 |

[7] | [Ji] Jimbo, M.: Aq-difference analogue ofU(G) and the Yang-Baxter equation. Lett. Math. Phys.10, 63-69 (1985) · Zbl 0587.17004 |

[8] | [Jo] Jones, V. F. R.: Polynomial invariants of knots via non Neumann algebras. Bull. Am. Math. Soc.12, 103-111 (1985) · Zbl 0564.57006 |

[9] | [Ka] Kac, V. G.: Infinite dimensional Lie algebras. (Progr. Math. vol. 44). Boston: Birkhäuser 1983; 2nd ed., Cambridge University Press, 1985 · Zbl 0537.17001 |

[10] | [KaW] Kac, V. G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. Math.40, 1, 156-236 (1988) · Zbl 0661.17016 |

[11] | [K] Kirby, R.: The calculus of framed links inS 3. Invent. Math.45, 35-56 (1978) · Zbl 0377.55001 |

[12] | [KiR] Kirillov, A.N., Reshetikhin, N.Yu: Representations of the algebraU q (sl 2),q-orthogonal polynomials and invariants of links. Leningrad: LOMI preprint E-9-88 1988 |

[13] | [KR] Kulish, P.P., Reshetikhin, N.Yu.: Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova101, 101-110 (in Russian) (1981) |

[14] | [Ki] Kirillov, A.N.: Quantum Clebsh-Gordan coefficients. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova168, 67-84 (1988) |

[15] | [Lu] Luztig, G.: Quantum deformations of certain simple modules over enveloping algebra 1987 (Preprint) |

[16] | [L] Lickorish, W.B.R.: A representation of orientable combinatorial 3-manifolds, Ann. Math.76, 531-540 (1962) · Zbl 0106.37102 |

[17] | [MR] Moore, G., Reshetikhin, N.: A comment on Quantum Group Symmetry in Conformal Field Theory. IAS preprint LASSNS-HEP-89/18 |

[18] | [MS] Moore, G., Seigerg, N.: Classical and quantum conformal field theories. IAS preprint IASSNS-HEP-88/39. |

[19] | [R] Rolfsen, D.: Knots and links. Boston Publish or Perish Press 1976 · Zbl 0339.55004 |

[20] | [Re] Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links. I., LOMI Preprint E-4-87, 1988; II. LOMI preprint E-17-87, 1988 |

[21] | [Re1] Reshetikhin, N.Yu.: Quasitriangular Hopf algebras and invariants of links. {jtAlgebra and Analysis}, v. {vn1}, {snN2}, {dy1989} |

[22] | [ReT] Reshetikhin, N.Yu., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys.,127, 1-26 (1990) · Zbl 0768.57003 |

[23] | [RT] Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. M.S.R.I. (Preprint April 1989) |

[24] | [Tu] Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math.92, 527-553 (1988) · Zbl 0648.57003 |

[25] | [Tu1] Turaev, V.G.: Operator invariants of tangles andR-matrices, Izv. AN SSSR,53, 1073-1107 (in Russian) (1989) |

[26] | [V] Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Physics B300, 360-380 (1988) · Zbl 1180.81120 |

[27] | [W] Wallace, A.H.: Modifications and cobounding manifolds. Can. J. Math.12, 503-528 (1960) · Zbl 0108.36101 |

[28] | [Wi] Witten, E.: Quantum field theory and Jones polynomial. Commun. Math. Phys.121 351-399 (1989) · Zbl 0667.57005 |

[29] | [Wi1] Witten, E.: Gauge Theories and Integrable Lattice Models. IAS IASSNS-HEP-89/11 (Preprint) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.