×

zbMATH — the first resource for mathematics

A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres. (English) Zbl 1144.57006
For integral homology \(3\)-spheres \(M\), the author introduces an invariant \(J_M\) which unifies the \(SU(2)\) Witten-Reshetikhin-Turaev (WRT) invariants at all roots of unity. It should be compared with the fact that the Ohtsuki series by T. Ohtsuki [Math. Proc. Camb. Philos. Soc. 117, No. 1, 83–112 (1995; Zbl 0843.57019)] unifies the WRT invariants at roots of unity of odd prime orders. This invariant was announced in a previous paper by the author [Geom. Topol. Monogr. 4, 55–68 (2002; Zbl 1040.57010)].
It takes values in a completion \(\widehat{\mathbb{Z}[q]}\) of the polynomial ring \(\mathbb{Z}[q]\), and its evaluation at any root of unity \(\zeta\) gives the WRT invariant \(\tau_\zeta(M)\). As an immediate consequence, \(\tau_\zeta(M)\in \mathbb{Z}[\zeta]\), which generalizes the integrality result by H. Murakami [Math. Proc. Camb. Philos. Soc. 115, No. 2, 253–281 (1994; Zbl 0832.57005)].
Let \(\mathcal{Z}\subset \mathbb{C}\) be the set of all roots of unity. For each \(M\), the WRT function \(\tau(M):\mathcal{Z}\to\mathbb{C}\) is defined by \(\tau(M)(\zeta)=\tau_\zeta(M)\). Then it is shown that \(J_M\) and \(\tau(M)\), also \(J_M\) and the Ohtsuki series \(\tau^O(M)\), have the same strength in distinguishing integral homology spheres.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Akutsu, Y., Deguchi, T., Ohtsuki, T.: Invariants of colored links. J. Knot Theory Ramifications 1(2), 161–184 (1992) · Zbl 0758.57004 · doi:10.1142/S0218216592000094
[2] Bar-Natan, D., Garoufalidis, S.: On the Melvin–Morton–Rozansky conjecture. Invent. Math. 125(1), 103–133 (1996) · Zbl 0855.57004 · doi:10.1007/s002220050070
[3] Beliakova, A., Blanchet, C., Le, T.: Laplace transform and universal sl(2) invariants. Preprint, math.QA/0509394
[4] Cochran, T.D., Melvin, P.: Finite type invariants of 3-manifolds. Invent. Math. 140, 45–100 (2000) · Zbl 0949.57010 · doi:10.1007/s002220000045
[5] Crane, L., Yetter, D.: On algebraic structures implicit in topological quantum field theories. J. Knot Theory Ramifications 8(2), 125–163 (1999) · Zbl 0935.57025 · doi:10.1142/S0218216599000109
[6] Fenn, R., Rourke, C.: On Kirby’s calculus of links. Topology 18(1), 1–15 (1979) · Zbl 0413.57006 · doi:10.1016/0040-9383(79)90010-7
[7] Garoufalidis, S., Lê, T.T.Q.: Is the Jones polynomial of a knot really a polynomial? J. Knot Theory Ramifications 15, 983–1000 (2006) · Zbl 1126.57007 · doi:10.1142/S0218216506004919
[8] Garoufalidis, S., Levine, J.: Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism. In: Graphs and Patterns in Mathematics and Theoretical Physics. Proc. Sympos. Pure Math., vol. 3, pp. 173–203. Am. Math. Soc., Providence, RI (2005) · Zbl 1086.57013
[9] Gilmer, P.M.: Integrality for TQFTs. Duke Math. J. 125(2), 389–413 (2004) · Zbl 1107.57020 · doi:10.1215/S0012-7094-04-12527-8
[10] Gilmer, P.M., Masbaum, G.: Integral lattices in TQFT. To appear in Ann. Sci. Éc. Norm. Supér., IV. Sér. · Zbl 1178.57023
[11] Gilmer, P.M., Masbaum, G., van Wamelen, P.: Integral bases for TQFT modules and unimodular representations of mapping class groups. Comment. Math. Helv. 79(2), 260–284 (2004) · Zbl 1055.57026 · doi:10.1007/s00014-004-0801-5
[12] Goussarov, M.: Finite type invariants and n-equivalence of 3-manifolds. C. R. Acad. Sci. Paris, Sèr. I 329(6), 517–522 (1999) · Zbl 0938.57013
[13] Gouvêa, F.Q.: p-adic numbers. An introduction, 2nd edn. Universitext. Springer, Berlin (1997) · Zbl 0874.11002
[14] Habegger, N.: Milnor, Johnson, and tree level perturbative invariants. Preprint
[15] Habiro, K.: Claspers and finite type invariants of links. Geom. Topol. 4, 1–83 (2000) · Zbl 0941.57015 · doi:10.2140/gt.2000.4.1
[16] Habiro, K.: On the colored Jones polynomials of some simple links. Recent progress towards the volume conjecture (Kyoto, 2000). RIMS Kokyuroku 1172, 34–43 (2000) · Zbl 0969.57503
[17] Habiro, K.: On the quantum sl 2 invariants of knots and integral homology spheres. In: Invariants of Knots and 3-manifolds (Kyoto 2001). Geom. Topol. Monogr., vol. 4, pp. 55–68 (2002) · Zbl 1040.57010
[18] Habiro, K.: Cyclotomic completions of polynomial rings. Publ. Res. Inst. Math. Sci. 40, 1127–1146 (2004) · Zbl 1098.13032 · doi:10.2977/prims/1145475444
[19] Habiro, K.: An integral form of the quantized enveloping algebra of sl 2 and its completions. J. Pure Appl. Algebra 211, 265–292 (2007) · Zbl 1194.17005 · doi:10.1016/j.jpaa.2007.01.011
[20] Habiro, K.: Bottom tangles and universal invariants. Algebr. Geom. Topol. 6, 1113–1214 (2006) · Zbl 1130.57014 · doi:10.2140/agt.2006.6.1113
[21] Habiro, K.: Refined Kirby calculus for integral homology spheres. Geom. Topol. 10, 1285–1317 (2006) · Zbl 1130.57028 · doi:10.2140/gt.2006.10.1285
[22] Habiro, K.: Spanning surfaces and the Jones polynomial. In preparation · Zbl 0969.57503
[23] Habiro, K., Le, T.T.Q.: In preparation
[24] Hikami, K.: Quantum invariant, modular form, and lattice points. Int. Math. Res. Not. 2005(3), 121–154 (2005) · Zbl 1075.57005 · doi:10.1155/IMRN.2005.121
[25] Hikami, K.: On the quantum invariant for the Brieskorn homology spheres. Int. J. Math. 16(6), 661–685 (2005) · Zbl 1088.57013 · doi:10.1142/S0129167X05003004
[26] Hikami, K.: Mock (false) theta functions as quantum invariants. Regul. Chaotic Dyn. 10(4), 509–530 (2005) · Zbl 1133.57301 · doi:10.1070/RD2005v010n04ABEH000328
[27] Hikami, K.: Quantum invariants, modular forms, and lattice points II. J. Math. Phys. 47, 102301, 32pp. (2006) · Zbl 1112.58026
[28] Hoste, J.: A formula for Casson’s invariant. Trans. Am. Math. Soc. 297(2), 547–562 (1986) · Zbl 0604.57008
[29] Huynh, V., Le, T.T.Q.: On the colored Jones polynomial and the Kashaev invariant. Fundam. Prikl. Mat. 11, 57–78 (2005)
[30] Jones, V.F.R.: A new knot polynomial and von Neumann algebras. Notices Am. Math. Soc. 33(2), 219–225 (1986)
[31] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126, 335–388 (1987) · Zbl 0631.57005 · doi:10.2307/1971403
[32] Kashaev, R.M.: A link invariant from quantum dilogarithm. Modern Phys. Lett. A 10, 1409–1418 (1995) · Zbl 1022.81574 · doi:10.1142/S0217732395001526
[33] Kauffman, L.H.: Gauss codes, quantum groups and ribbon Hopf algebras. Rev. Math. Phys. 5(4), 735–773 (1993) · Zbl 0803.57001 · doi:10.1142/S0129055X93000231
[34] Kauffman, L., Radford, D.E.: Oriented quantum algebras, categories and invariants of knots and links. J. Knot Theory Ramifications 10(7), 1047–1084 (2001) · Zbl 1008.57010 · doi:10.1142/S0218216501001268
[35] Kerler, T.: Genealogy of non-perturbative quantum invariants of 3–manifolds: The surgical family. In: Geometry and Physics (Aarhus, 1995). Lect. Notes Pure Appl. Math., vol. 184, pp. 503–547. Dekker, New York (1997) · Zbl 0869.57014
[36] Kerler, T.: Bridged links and tangle presentations of cobordism categories. Adv. Math. 141(2), 207–281 (1999) · Zbl 0937.57017 · doi:10.1006/aima.1998.1772
[37] Kirby, R.: A calculus for framed links in S 3. Invent. Math. 45(1), 35–56 (1978) · Zbl 0377.55001 · doi:10.1007/BF01406222
[38] Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2,C). Invent. Math. 105(3), 473–545 (1991) · Zbl 0745.57006 · doi:10.1007/BF01232277
[39] Kirby, R., Melvin, P., Zhang, X.: Quantum invariants at the sixth root of unity. Commun. Math. Phys. 151(3), 607–617 (1993) · Zbl 0779.57007 · doi:10.1007/BF02097030
[40] Koblitz, N.: p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, 2nd edn. Grad. Texts Math., vol. 58. Springer, New York (1984) · Zbl 0364.12015
[41] Kricker, A., Spence, B.: Ohtsuki’s invariants are of finite type. J. Knot Theory Ramifications 6(4), 583–597 (1997) · Zbl 0885.57006 · doi:10.1142/S0218216597000327
[42] Lawrence, R.J.: A universal link invariant using quantum groups. In: Differential Geometric Methods in Theoretical Physics (Chester, 1988), pp. 55–63. World Sci. Publishing, Teaneck, NJ (1989)
[43] Lawrence, R.J.: A universal link invariant. In: The Interface of Mathematics and Particle Physics (Oxford, 1988). Inst. Math. Appl. Conf. Ser. New Ser., vol. 24, pp. 151–156. Oxford University Press, New York (1990)
[44] Lawrence, R.J.: Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3-manifolds. J. Math. Phys. 36, 6106–6129 (1995) · Zbl 0877.57008 · doi:10.1063/1.531237
[45] Lawrence, R.J.: Witten–Reshetikhin–Turaev invariants of 3-manifolds as holomorphic functions. In: Geometry and Physics (Aarhus, 1995). Lect. Notes Pure Appl. Math., vol. 184, pp. 363–377. Dekker, New York (1997) · Zbl 0905.57012
[46] Lawrence, R., Rozansky, L.: Witten–Reshetikhin–Turaev invariants of Seifert manifolds. Commun. Math. Phys. 205(2), 287–314 (1999) · Zbl 0966.57017 · doi:10.1007/s002200050678
[47] Lawrence, R., Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. Math. 3, 93–107 (1999) · Zbl 1024.11028
[48] Le, T.T.Q.: An invariant of integral homology 3-spheres which is universal for all finite type invariants. In: Solitons, Geometry, and Topology: On the Crossroad. Trans. Am. Math. Soc., vol. 179, pp. 75–100. Am. Math. Soc., Providence, RI (1997) · Zbl 0914.57013
[49] Le, T.T.Q.: On perturbative PSU(n) invariants of rational homology 3-spheres. Topology 39(4), 813–849 (2000) · Zbl 0955.57010 · doi:10.1016/S0040-9383(99)00037-3
[50] Le, T.T.Q.: Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. Topol. Appl. 127(1–2), 125–152 (2003) · Zbl 1020.57002 · doi:10.1016/S0166-8641(02)00056-1
[51] Le, T.T.Q.: Strong integrality of quantum invariants of 3-manifolds. To appear in Trans. Am. Math. Soc.
[52] Le, T.T.Q., Murakami, J., Ohtsuki, T.: On a universal perturbative invariant of 3-manifolds. Topology 37(3), 539–574 (1998) · Zbl 0897.57017 · doi:10.1016/S0040-9383(97)00035-9
[53] Lee, H.C.: Tangles, links, and twisted quantum groups. In: Physics, Geometry and Topology (Banff, 1989). NATO ASI Ser., Ser. B, Phys., vol. 238, pp. 623–655. Plenum, New York (1990) · Zbl 0748.57001
[54] Lin, X., Wang, Z.: On Ohtsuki’s invariants of integral homology 3-spheres. Acta Math. Sin. 15(3), 293–316 (1999) · Zbl 0936.57011 · doi:10.1007/BF02650726
[55] Lin, X.-S., Wang, Z.: Fermat limit and congruence of Ohtsuki invariants. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998). Geom. Topol. Monogr., vol. 2, pp. 321–333. Geom. Topol. Publ., Coventry (1999) · Zbl 0959.57008
[56] Majid, S.: Algebras and Hopf algebras in braided categories. In: Advances in Hopf Algebras. Lect. Notes Pure Appl. Math., vol. 158, pp. 55–105. Dekker, New York (1994) · Zbl 0812.18004
[57] Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995) · Zbl 0857.17009
[58] Masbaum, G.: Skein-theoretical derivation of some formulas of Habiro. Algebr. Geom. Topol. 3, 537–556 (2003) · Zbl 1042.57005 · doi:10.2140/agt.2003.3.537
[59] Masbaum, G., Roberts, J.D.: A simple proof of integrality of quantum invariants at prime roots of unity. Math. Proc. Camb. Philos. Soc. 121(3), 443–454 (1997) · Zbl 0882.57010 · doi:10.1017/S0305004196001624
[60] Masbaum, G., Wenzl, H.: Integral modular categories and integrality of quantum invariants at roots of unity of prime order. J. Reine Angew. Math. 505, 209–235 (1998) · Zbl 0919.57010 · doi:10.1515/crll.1998.505.209
[61] Melvin, P.M., Morton, H.R.: The coloured Jones function. Commun. Math. Phys. 169, 501–520 (1995) · Zbl 0845.57007 · doi:10.1007/BF02099310
[62] Murakami, H.: Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant. Math. Proc. Camb. Philos. Soc. 115(2), 253–281 (1994) · Zbl 0832.57005 · doi:10.1017/S0305004100072078
[63] Murakami, H.: Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson and Walker. Math. Proc. Camb. Philos. Soc. 117(2), 237–249 (1995) · Zbl 0854.57016 · doi:10.1017/S0305004100073084
[64] Murakami, H., Murakami, J.: The coloured Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001) · Zbl 0983.57009 · doi:10.1007/BF02392716
[65] Murakami, J.: A state model for the multi-variable Alexander polynomial. Pac. J. Math. 157, 109–135 (1993) · Zbl 0799.57006
[66] Ohtsuki, T.: Colored ribbon Hopf algebras and universal invariants of framed links. J. Knot Theory Ramifications 2(2), 211–232 (1993) · Zbl 0798.57006 · doi:10.1142/S0218216593000131
[67] Ohtsuki, T.: A polynomial invariant of integral homology 3-spheres. Math. Proc. Camb. Philos. Soc. 117(1), 83–112 (1995) · Zbl 0843.57019 · doi:10.1017/S0305004100072935
[68] Ohtsuki, T.: A polynomial invariant of rational homology 3-spheres. Invent. Math. 123(2), 241–257 (1996) · Zbl 0855.57016
[69] Ohtsuki, T.: Finite type invariants of integral homology 3-spheres. J. Knot Theory Ramifications 5(1), 101–115 (1996) · Zbl 0942.57009 · doi:10.1142/S0218216596000084
[70] Ohtsuki, T.: The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant. Topology 39, 1103–1135 (2000) · Zbl 1006.57006 · doi:10.1016/S0040-9383(99)00001-4
[71] Ohtsuki, T. (ed.): Problems on invariants of knots and 3-manifolds. In: Invariants of Knots and 3-Manifolds (Kyoto 2001). Geom. Topol. Monogr., vol. 4, pp. 377–572. Geom. Topol. Publ., Coventry (2002) · Zbl 1163.57302
[72] Reshetikhin, N.Y.: Quasitriangle Hopf algebras and invariants of tangles. Algebr. Anal. 1(2), 169–188 (1989) (Russian) (English transl.: Leningr. Math. J. 1(2), 491–513 (1990))
[73] Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991) · Zbl 0725.57007 · doi:10.1007/BF01239527
[74] Rozansky, L.: The universal R-matrix, Burau representation, and the Melvin–Morton expansion of the colored Jones polynomial. Adv. Math. 134, 1–31 (1998) · Zbl 0949.57006 · doi:10.1006/aima.1997.1661
[75] Rozansky, L.: On p-adic properties of the Witten–Reshetikhin–Turaev invariant. In: Primes and Knots. Contemp. Math., vol. 416, pp. 213–236. Am. Math. Soc., Providence, RI (2006) · Zbl 1137.57018
[76] Takata, T., Yokota, Y.: The PSU(N) invariants of 3-manifolds are algebraic integers. J. Knot Theory Ramifications 8(4), 521–532 (1999) · Zbl 0933.57019 · doi:10.1142/S0218216599000365
[77] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
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.