Thang T. Q. Le

( Tự Quốc Thắng)


Thang Le

School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
Tel:     (404)385-0054
FAX:  (404)894-4409



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

+        Recent preprints, papers, talks, etc...



+          Please visit Canvas for course syllabus and information.




+        Managing 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.

New: I am co-organizing Conference “Quantum Topology and Hyperbolic Geometry Conference Da Nang, Vietnam during May 27-31, 2019. Following the conference, there will be a workshop consisting of 3 min-courses held at Quy Nhon University.



Research papers


  1. Quantum traces and embeddings of stated skein algebras into quantum tori, with T. Yu. Preprint arXiv:2012.15272. Link to pdf.
  2. The Chebyshev-Frobenius Homomorphism for stated skein modules of 3-manifolds, with W. Bloomquist, Preprint arXiv:2011.02130.  link to Pdf
  3. SL_2 quantum trace in quantum Teichmüller theory via writhe, With H. K. Kim and M. Son, Preprint Arxiv:1812.11628. (new) Pdf file.
  4. Stated skein modules of marked 3-manifolds/surfaces, a survey, With T. Yu, Preprint arXiv:2005.14577, 2020.  pdf
  5. Lower and upper bounds for positive bases of skein algebras. With D. Thurston and T. Yu. International Mathematics Research Notices, to appear. Preprint arXiv:1908.05775, 2019. pdf
  6. Stated skein algebras of surfaces. With F. Costantino. Journal of EMS, to appear. Preprint arXiv:1907.11400, 2019. New version pdf.
  7. Dimension and Trace of the Kauffman Bracket Skein Algebra, With C. Frohman and J. Kania-Bartoszynska. Preprint arXiv:1902.02002, 2019. pdf
  8. 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.
  9. Unicity for Representations of the Kauffman bracket Skein Algebra. With C. Frohman and J. Kania-Bartoszynska. Invent. Math. 215 (2019), no. 2, 609-650. Here is a pdf.
  10. Triangular decomposition of skein algebras,   Quantum Topology, 9 (2018), 591-632. pdf.
  11. The colored Homflypt function is q-holonomic. With S. Garoufalidis and A. Lauda. pdf. Duke Math. Journal, 167, Number 3 (2018), 397-447.
  12. On positivity of Kauffman bracket skein algebras of surfaces, International Mathematics Research Notices, 2018, no. 5, 1314-1328. pdf.
  13. 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.
  14. 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.
  15. 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
  16. Character varieties, A-polynomials, and the AJ Conjecture. With Xingru Zhang. pdf. Algebraic & Geometric Topology, 17 (2017), no. 1, 157-188.
  17. 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.
  18. Growth of  homology torsion in finite coverings and hyperbolic volume. Preprint arXiv:1412.7758, 2014. pdf. Annales de l'Institut Fourier, 68 (2018), no. 2, 611--645.
  19. On Kauffman Bracket Skein Modules at Root of Unity. Algebraic & Geometric Topology,  15 (2015), no. 2, 1093-1117. pdf
  20. Nahm sums, stability and the colored Jones polynomial. With S. Garoufalidis.  Research in Mathematical Sciences, 2 (2015), Art. 1, 55 pp.  pdf
  21. On the AJ conjecture for knots. With Anh Tran. Indiana Univ. Math. J. 64 (2015), no. 4, 1103-1151.   pdf
  22. Homology torsion growth and Mahler measure. Comment. Math. Helv 89 (2014), no. 3, 719-757. pdf
  23. On the integrality of the Witten-Reshetikhin-Turaev 3-manifold invariants. With A. Beliakova and Q. Chen. Quantum Topology  5 (2014), 99-141  .pdf
  24. The skein module of two-bridge links. With Anh Tran. Proceedings  Amer. Math. Soc. 142 (2014), 1045-1056. pdf
  25. Growth of regulators in finite abelian coverings. Algebraic & Geometric Topology 13 (2013), 2383-2404. pdf
  26. 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
  27. 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
  28. Asymptotics of the colored Jones function of a knot. With S. Garoufalidis. Geometry and Topology,  15 (2011), 2135-2180. .pdf
  29. Unified SO(3) Quantum Invariants for Rational Homology 3-Spheres. With A. Beliakova and I. Buehler, Invent. Math. 185 (2011), 121-174.  .pdf
  30. On the Volume Conjecture for Cables of Knots. With Anh Tran. J. Knot Theory Ramifications, 19 (2010),  1673--1691. .pdf
  31. 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
  32. 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
  33. Strong Integrality of Quantum Invariants of 3-manifolds.  Trans. Amer. Math. Soc.  360  (2008),  2941-2963.  .pdf
  34. Twisted Alexander Polynomial of links in the projective space. With Vu Huynh. Knot Theory Ramifications  17  (2008), 411-438.   pdf
  35. Gevrey series in quantum topologyWith S. Garoufalidis. J. Reine Angew. Math.  618  (2008), 169-195.  .pdf
  36. Integrality of quantum 3-manifold invariants and a rational surgery formula. With A. Beliakova. Compositio Mathematica, 143 (2007), 1593-1612.  .pdf
  37. A TQFT associated to the LMO invariant of three-dimensional manifolds. With D. Cheptea. Comm. Math. Phys.  272  (2007),   601-634. pdf
  38. Is the Jones polynomial of a knot really a polynomial? With S. Garoufalidis. J. Knot Theory and its Ramification 15 (2006), 983--1000.  pdf 
  39. The Colored Jones Polynomial and the A-Polynomial of Knots. Adv. in Math. 207 (2006), 782--804   .pdf . Corrections.
  40. The quantum MacMahon Master Theorem. With S. Garoufalidis and  D. Zeiberger. Proceedings of the Nat. Acad. Sci. 103 (2006), N 38, 13928--13931. pdf.
  41. On the Colored Jones Polynomial and the Kashaev invariant. With Vu Huynh. Fundam. Prikl. Mat.  11  (2005),  no. 5, 57-78.  pdf
  42. Almost integral TQFTs from simple Lie algebras. With Q. Chen. Algebr. Geom. Topol.  5 (2005), 1291--1314.  pdf
  43. The colored Jones function is q-holonomic. With S. Garoufalidis. Geom. Topol.  9  (2005), 1253--1293.  pdf
  44. Quantum Invariants of Periodic Links and Periodic 3-Manifolds. With Q. Chen. Fund. Math.  184  (2004), 55--71.  pdf
  45. Quantum groups and ribbon G-categories. With V. Turaev. J. Applied and Pure Algebra  178 (2003), 169--185.  pdf
  46. 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
  47. Quantum 3-manifolds invariants: integrality, splitting, and perturbative expansion. Topology and its applications, 127  (2003), 125-152.  .pdf
  48. Integrality and symmetry of quantum link invariants. Duke Math. J. 102 (2000), no. 2, 273--306.  pdf 
  49. On the perturbative PSU(n) invariants of rational homology 3-spheres. Topology 39 (2000), no. 4, 813-849.  .pdf
  50. 3-manifold invariant derived from the Kontsevich integral. With J. Murakami, H. Murakami, T. Ohtsuki. Osaka J. Math. 36 (1999), no. 2, 365--395.  
  51. On the  denominators of the Kontsevich integral and the universal perturbative invariant. Invent. Math. 135 (1999), no. 3, 689-722.  .pdf
  52. A universal perturbative invariant of 3-manifolds. With J. Murakami and T. Ohtsuki. Topology, 37 (1998), 539--574.   pdf 
  53. The parallel version of the Kontsevich integral. With J. Murakami.J.  Pure and Applied Algebra, 121 (1997), 271--291.  .pdf
  54. 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
  55. 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
  56. 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
  57. The Universal Vassiliev-Kontsevich Invariant for framed oriented links. With J. Murakami. Compositio Mathematics 102 (1996), pp. 41--64.  pdf
  58. The Kontsevich integral  for Kauffman's polynomials. With J. Murakami. Nagoya Mathematical Journal, {\bf 142} (1996), pp. 39--66.  
  59. 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. 
  60. Representation of the  category of Tangles by Kontsevich  iterated integrals. With J. Murakami. Commun.  Math.  Physics 168 (1995), 535-562.  .pdf
  61. Kontsevich's integrals for Homfly polynomials and relations between multiple zeta values. With J. Murakami. Topology and its Applications, 62 (1995), 193--206. pdf
  62. Local rules for pentagonal quasiperiodic tilings Discrete  and  Computational Geometry,  14 (1995), 31-70. pdf 
  63. Local rules for multidimensional quasiperiodic tilings. With S. Piunikhin. Differential  Geometry and its applications, 5 (1995), 13-31.  pdf 
  64. 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.
  65. 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
  66. 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
  67. 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.  
  68. Novikov's numbers. (Russian) Mat. Zametki  47 (1990), no. 1, 98-104; translation in  Math. Notes 47 (1990), no. 1-2, 64-68.  
  69. 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

  1. Finite type invariants of 3-manifolds. In ``Encyclopedia of Mathematical Physics", pp 384-356, Elsevier 2006.  pdf
  2. Lectures on the LMO invariant. Lectures given at the Grenoble 1999 Summer School on invariants of 3-manifolds. pdf 
  3. Aktueller Forschungsschwerpunkt: Invarianten von Knoten. With M. Kontsevich. In ``Max-Planck-Gesellschaft Jahrbuch 1994", pp 451--455. Max-Planck-Gesellschaft. pdf