Thang T. Q. Le

(Lê Tự Quốc Thắng)


Thang Le

School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
FAX:  (404)894-4409



Research Interests: Differential topology, 3-manifolds, knot theory, quasicrystals.



+          Please visit Canvas for course syllabus and information.




+        Editor, Quantum TopologyPapers are submitted to The European Mathemathical Society Publishing House at this site.

+        Editor for The Journal of Knot theory and its Ramifications. Papers are submitted to World Scientific at this site.

+        Editor for Acta Mathematica Vietnamica.  Papers are submitted to Springer at this site.




Research papers



1.  Sliced skein algebras and geometric Kauffman bracket, with C. Frohman and J. Kania-Bartoszynska, arXiv:2310.06189

2.  Degenerations Of Skein Algebras And Quantum Traces, with W. Bloomquist and H. Karuo, preprint arXiv:2308.16702.


3.      From 3-dimensional skein theory to functions near Q, with S. Garoufalidis,

4.     Quantum traces for SL_n-skein algebras, with T. Yu, Preprint ArXiv:2303.08082

5.     Stated skein modules of 3-manifolds and TQFT, With F. Costantino, prerint arXiv:2206.10906

6.     Stated SL(n)-Skein Modules and Algebras, with A. Sikora, Preprint arXiv 2201.00045.

7.     Root of unity quantum cluster algebras and Cayley-Hamilton algebras, with S Huang and M. Yakimov. Trans AMS, to appear. Preprint arXiv:2107.11926.

8.     Faithfulness of geometric action of skein algebras. Acta Math. Vietnam. 47 (2022), no. 1, 279--291. Preprint arXiv:2103.11532

9.     Quantum traces and embeddings of stated skein algebras into quantum tori, with T. Yu. Selecta Math. (N.S.) 28 (2022), no. 4, Paper No. 66, 48 pp. Preprint arXiv:2012.15272.

10.  The Chebyshev-Frobenius Homomorphism for stated skein modules of 3-manifolds, with W. Bloomquist, Math. Z. 301 (2022), no. 1, 1063-1105. Preprint arXiv:2011.02130.

11.  SL_2 quantum trace in quantum Teichmüller theory via writhe, With H. K. Kim and M. Son,  Algebr. Geom. Topol, to appear. Preprint Arxiv:1812.11628. (new, updated version).

12.  Stated skein modules of marked 3-manifolds/surfaces, a survey, With T. Yu, Acta Math. Vietnam. 46 (2021), no. 2, 265–287. pdf

13.  Lower and upper bounds for positive bases of skein algebras. With D. Thurston and T. Yu. Int. Math. Res. Not. IMRN 2021, no. 4, 3186-3202. Link to pdffile

14.  Stated skein algebras of surfaces. With F. Costantino. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 12, 4063--4142. Preprint arXiv:1907.11400.

15.  Dimension and Trace of the Kauffman Bracket Skein Algebra, With C. Frohman and J. Kania-Bartoszynska. Trans. Amer. Math. Soc. Ser. B 8 (2021), 510–547. Link to pdf.

16.  On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism. With J. Paprocki. Algebr. Geom. Topol. 19 (2019), no. 7, 3453-3509. Preprint arXiv:1804.09303, 2018. pdf.

17.  Unicity for Representations of the Kauffman bracket Skein Algebra. With C. Frohman and J. Kania-Bartoszynska. Invent. Math. 215 (2019), no. 2, 609-650.  pdf.

18.  Triangular decomposition of skein algebras,   Quantum Topology, 9 (2018), 591-632. pdf.

19.  The colored Homflypt function is q-holonomic. With S. Garoufalidis and A. Lauda. pdf. Duke Math. Journal, 167, Number 3 (2018), 397-447.

20.  On positivity of Kauffman bracket skein algebras of surfaces, International Mathematics Research Notices, 2018, no. 5, 1314-1328. pdf.

21.  A survey of q-holonomic functions. With S. Garoufalidis. Preprint arXiv:1601.07487, 2016, pdf. L'Enseignement Mathematique, 62 (2016), no. 3-4, 501-525.

22.  Quantum Teichmuller spaces and quantum trace map, preprint  arXiv:1511.06054, 2015, pdf. Journal of the Institute of Mathematics of Jussieu, 18 (2019), no. 2, 249–291.

23.  Unified quantum invariants for integral homology spheres associated with simple Lie algebras. With K. Habiro. Geometry and Topology 20 (2016), 2687-2835. DOI 10.2140/gt.2016.20.2687.  pdf

24.  Character varieties, A-polynomials, and the AJ Conjecture. With Xingru Zhang. pdf. Algebraic & Geometric Topology, 17 (2017), no. 1, 157-188.

25.  The colored Jones polynomial and the AJ conjecture. Lecture notes from "Etats de la Recherche : Geometric and quantum topology in dimension 3" at the CIRM in Luminy, July 2014. pdf. Appeared in in ``Lectures on quantum topology in dimension three" (by T. Le, C. Lescop, R. Lipshitz, P. Turner), Panoramas et Syntheses, N 48 (2016), Soc. Math. France, pp 33-90.

26.  Growth of  homology torsion in finite coverings and hyperbolic volume. Annales de l'Institut Fourier, 68 (2018), no. 2, 611--645. Preprint arXiv:1412.7758.

27.  On Kauffman Bracket Skein Modules at Root of Unity. Algebraic & Geometric Topology, 15 (2015), no. 2, 1093-1117. pdf

28.  Nahm sums, stability and the colored Jones polynomial. With S. Garoufalidis.  Research in Mathematical Sciences, 2 (2015), Art. 1, 55 pp.  pdf

29.  On the AJ conjecture for knots. With Anh Tran. Indiana Univ. Math. J. 64 (2015), no. 4, 1103-1151.  pdf

30.  Homology torsion growth and Mahler measure. Comment. Math. Helv.  89 (2014), no. 3, 719-757. pdf

31.  On the integrality of the Witten-Reshetikhin-Turaev 3-manifold invariants. With A. Beliakova and Q. Chen. Quantum Topology  5 (2014), 99-141  .pdf

32.  The skein module of two-bridge links. With Anh Tran. Proceedings  Amer. Math. Soc. 142 (2014), 1045-1056. pdf

33.  Growth of regulators in finite abelian coverings. Algebraic & Geometric Topology 13 (2013), 2383-2404. pdf

34.  The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant. With T. Kuriya and  T. Ohtsuki. J. Topology 5  (2012), 458--484.  pdf

35.  On the unification of quantum 3-manifold invariantsWith A. Beliakova,  in "Introductory lectures on knot theory",  Ser. Knots Everything 46, World Sci. Publ., Hackensack, NJ, 2012, 1--21,  .pdf

36.  Asymptotics of the colored Jones function of a knot. With S. Garoufalidis. Geometry and Topology,  15 (2011), 2135-2180. .pdf

37.  Unified SO(3) Quantum Invariants for Rational Homology 3-Spheres. With A. Beliakova and I. Buehler, Invent. Math. 185 (2011), 121-174.  .pdf

38.  On the Volume Conjecture for Cables of Knots. With Anh Tran. J. Knot Theory Ramifications, 19 (2010),  1673--1691. .pdf

39.  Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion. With A. Beliakova and C. Blanchet. Fund. Math.  201  (2008),  217-239.  .pdf

40.  Analyticity of the free energy of a closed 3-manifold. With S. Garoufalidis and M. Marino. Symmetry, Integrability and Geometry: Methods and Applications, 4 (2008), 080, 20 pages (electronic). .pdf

41.  Strong Integrality of Quantum Invariants of 3-manifolds.  Trans. Amer. Math. Soc.  360  (2008),  2941-2963.  .pdf

42.  Twisted Alexander Polynomial of links in the projective space. With Vu Huynh. Knot Theory Ramifications  17  (2008), 411-438.   pdf

43.  Gevrey series in quantum topologyWith S. Garoufalidis. J. Reine Angew. Math.  618  (2008), 169-195.  .pdf

44.  Integrality of quantum 3-manifold invariants and a rational surgery formula. With A. Beliakova. Compositio Mathematica, 143 (2007), 1593-1612.  .pdf

45.  A TQFT associated to the LMO invariant of three-dimensional manifolds. With D. Cheptea. Comm. Math. Phys.  272  (2007),   601-634. pdf

46.  Is the Jones polynomial of a knot really a polynomial? With S. Garoufalidis. J. Knot Theory and its Ramification 15 (2006), 983--1000.  pdf 

47.  The Colored Jones Polynomial and the A-Polynomial of Knots. Adv. in Math. 207 (2006), 782--804   .pdf . Corrections.

48.  The quantum MacMahon Master Theorem. With S. Garoufalidis and  D. Zeiberger. Proceedings of the Nat. Acad. Sci. 103 (2006), N 38, 13928--13931. pdf.

49.  On the Colored Jones Polynomial and the Kashaev invariant. With Vu Huynh. Fundam. Prikl. Mat.  11  (2005),  no. 5, 57-78.  pdf

50.  Almost integral TQFTs from simple Lie algebras. With Q. Chen. Algebr. Geom. Topol.  5 (2005), 1291--1314.  pdf

51.  The colored Jones function is q-holonomic. With S. Garoufalidis. Geom. Topol.  9  (2005), 1253--1293.  pdf

52.  Quantum Invariants of Periodic Links and Periodic 3-Manifolds. With Q. Chen. Fund. Math.  184  (2004), 55--71.  pdf

53.  Quantum groups and ribbon G-categories. With V. Turaev. J. Applied and Pure Algebra  178 (2003), 169--185.  pdf

54.  Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. With D. Bar-Natan and D. Thurston. Geometry and Topology, 7 (2003), 1-31.  pdf

55.  Quantum 3-manifolds invariants: integrality, splitting, and perturbative expansion. Topology and its applications, 127  (2003), 125-152.  .pdf

56.  Integrality and symmetry of quantum link invariants. Duke Math. J. 102 (2000), no. 2, 273--306.  pdf 

57.  On the perturbative PSU(n) invariants of rational homology 3-spheres. Topology 39 (2000), no. 4, 813-849.  .pdf

58.  3-manifold invariant derived from the Kontsevich integral. With J. Murakami, H. Murakami, T. Ohtsuki. Osaka J. Math. 36 (1999), no. 2, 365--395.  

59.  On the  denominators of the Kontsevich integral and the universal perturbative invariant. Invent. Math. 135 (1999), no. 3, 689-722.  .pdf

60.  A universal perturbative invariant of 3-manifolds. With J. Murakami and T. Ohtsuki. Topology, 37 (1998), 539--574.   pdf 

61.  The parallel version of the Kontsevich integral. With J. Murakami.J.  Pure and Applied Algebra, 121 (1997), 271--291.  .pdf

62.  An invariant of integral homology 3-spheres which is universal for all finite types invariants.  In ``Soliton, Geometry, and Topology: On the Crossroad'', AMS Translations series 2, 179 (1997) Eds. V. Buchstaber and S. Novikov, pp. 75-100.  .pdf

63.  Local rules for quasiperiodic tilings. In "The mathematics of long-range aperiodic order", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.  489, Kluwer Acad. Publ., Dordrecht, 1997. pp 331-366. pdf

64.  A quantum $sl_2$-invariant of 3-manifolds which contains all the Witten-Reshetikhin-Turaev invariants. Lect. Notes in Pure and Appl. Math. 184, Marcel Dekker Inc. 1996. Eds. J. E. Andersen, J. Dupont, H. Pedersen and  A.Swann, 399-409.  .pdf

65.  The Universal Vassiliev-Kontsevich Invariant for framed oriented links. With J. Murakami. Compositio Mathematics 102 (1996), pp. 41--64.  pdf

66.  The Kontsevich integral  for Kauffman's polynomials. With J. Murakami. Nagoya Mathematical Journal, {\bf 142} (1996), pp. 39--66.  

67.  A 3-manifold invariant derived from the universal Vassiliev-Kontsevich invariant. With J. Murakami, H. Murakami, T. Otsuki. Proceeding of the  Japan Academy,  series A, Mathematical Sciences LXXI (1995), ser. A, N 6, pp. 125--127. 

68.  Representation of the  category of Tangles by Kontsevich  iterated integrals. With J. Murakami. Commun.  Math.  Physics 168 (1995), 535-562.  .pdf

69.  Kontsevich's integrals for Homfly polynomials and relations between multiple zeta values. With J. Murakami. Topology and its Applications, 62 (1995), 193--206. pdf

70.  Local rules for pentagonal quasiperiodic tilings Discrete  and  Computational Geometry,  14 (1995), 31-70. pdf 

71.  Local rules for multidimensional quasiperiodic tilings. With S. Piunikhin. Differential  Geometry and its applications, 5 (1995), 13-31.  pdf 

72.  The  geometry of quasicrystals.  With S. Piunikhin and V. Sadov.  (Russian) Uspekhi Mat. Nauk 48 (1993) , 41-102; translation in Russian Math. Surveys 48 (1993), no. 1, 37-100.

73.  Varieties of representations and their subvarieties of cohomology jumps for knot groups. (Russian) Mat. Sb. 184 (1993), no. 2, 57--82; translation in Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 187-209.  pdf

74.  Local rules for quasiperiodic tilings of quadratic 2-planes in R^4.  With S. Piunikhin and V. Sadov. Commun. Math. Phys. 150 (1992), 23-44.  pdf

75.  Varieties of representations and their subvarieties of cohomology jumps for certain knot groups.  (Russian) Uspekhi Mat. Nauk 46 (1991), no. 2(278), 223-224; translation in Russian Math. Surveys 46 (1991), no. 2, 250--251.  

76.  Novikov's numbers. (Russian) Mat. Zametki  47 (1990), no. 1, 98-104; translation in  Math. Notes 47 (1990), no. 1-2, 64-68.  

77.  Structure of level surfaces of closed 1-forms. (Russian) Mat. Zametki 44 (1988), no. 1, 124--133; translation in Math. Notes 44 (1988), no. 1-2, 556-562.

a.     Others

78.  Finite type invariants of 3-manifolds. In ``Encyclopedia of Mathematical Physics", pp 384-356, Elsevier 2006.  pdf

79.  Lectures on the LMO invariant. Lectures given at the Grenoble 1999 Summer School on invariants of 3-manifolds. pdf 

80.  Aktueller Forschungsschwerpunkt: Invarianten von Knoten. With M. Kontsevich. In ``Max-Planck-Gesellschaft Jahrbuch 1994", pp 451--455. Max-Planck-Gesellschaft. pdf