Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 025, 37 pages      arXiv:2008.06983      https://doi.org/10.3842/SIGMA.2026.025

A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$

Matthew Harper
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Received July 28, 2025, in final form February 18, 2026; Published online March 17, 2026

Abstract
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $\Delta_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $\Delta_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $\Delta_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $\Delta_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.

Key words: knots; quantum invariants; quantum groups at roots of unity; knot mutation.

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References

  1. Akutsu Y., Deguchi T., Ohtsuki T., Invariants of colored links, J. Knot Theory Ramifications 1 (1992), 161-184.
  2. Anghel C.A.-M., ADO invariants directly from partial traces of homological representations, New York J. Math. 30 (2024), 481-501, arXiv:2007.15616.
  3. Bar-Natan D., van der Veen R., Perturbed Gaussian generating functions for universal knot invariants, arXiv:2109.02057.
  4. Bar-Natan D., van der Veen R., A perturbed-Alexander invariant, Quantum Topol. 15 (2024), 449-472, arXiv:2206.12298.
  5. Blanchet C., Costantino F., Geer N., Patureau-Mirand B., Non-semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants, Adv. Math. 301 (2016), 1-78, arXiv:1404.7289.
  6. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  7. Cochran T.D., Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004), 347-398.
  8. Costantino F., Geer N., Patureau-Mirand B., Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories, J. Topol. 7 (2014), 1005-1053, arXiv:1202.3553.
  9. Costantino F., Gukov S., Putrov P., Non-semisimple TQFT's and BPS $q$-series, SIGMA 19 (2023), 010, 71 pages, arXiv:2107.14238.
  10. Costantino F., Harper M., Robertson A., Young M.B., Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$, Lett. Math. Phys. 116 (2026), 8, 48 page, arXiv:2407.12181.
  11. Creutzig T., Rupert M., Uprolling unrolled quantum groups, Commun. Contemp. Math. 24 (2022), 2150023, 27 pages, arXiv:2005.12445.
  12. De Concini C., Kac V.G., Representations of quantum groups at roots of $1$, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser, Boston, MA, 1990, 471-506.
  13. De Concini C., Kac V.G., Representations of quantum groups at roots of $1$: reduction to the exceptional case, in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 141-149.
  14. De Concini C., Kac V.G., Procesi C., Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151-189.
  15. De Wit D., An infinite suite of Links-Gould invariants, J. Knot Theory Ramifications 10 (2001), 37-62, arXiv:math.GT/0004170.
  16. De Wit D., Ishii A., Links J., Infinitely many two-variable generalisations of the Alexander-Conway polynomial, Algebr. Geom. Topol. 5 (2005), 405-418.
  17. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr., Vol. 205, American Mathematical Society, Providence, RI, 2015.
  18. Ferrari F., Putrov P., Supergroups, $q$-series and 3-manifolds, Ann. Henri Poincaré 25 (2024), 2781-2837, arXiv:2009.14196.
  19. Freyd P., Yetter D., Hoste J., Lickorish W.B.R., Millett K., Ocneanu A., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 239-246.
  20. Garoufalidis S., Harper M., Kohli B.M., Song J., Tahar G., Extending knot polynomials of braided Hopf algebras to links, arXiv:2505.01398.
  21. Garoufalidis S., Kashaev R., Multivariable knot polynomials from braided Hopf algebras with automorphisms, Publ. Res. Inst. Math. Sci. 62 (2026), 75-114, arXiv:2311.11528.
  22. Geer N., Patureau-Mirand B., Multivariable link invariants arising from $\mathfrak{sl}(2|1)$ and the Alexander polynomial, J. Pure Appl. Algebra 210 (2007), 283-298, arXiv:math.GT/0601291.
  23. Geer N., Patureau-Mirand B., Topological invariants from nonrestricted quantum groups, Algebr. Geom. Topol. 13 (2013), 3305-3363, arXiv:1009.4120.
  24. Geer N., Patureau-Mirand B., The trace on projective representations of quantum groups, Lett. Math. Phys. 108 (2018), 117-140, arXiv:1610.09129.
  25. Geer N., Patureau-Mirand B., Turaev V., Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 (2009), 196-212, arXiv:0711.4229.
  26. Gukov S., Pei D., Putrov P., Vafa C., BPS spectra and 3-manifold invariants, J. Knot Theory Ramifications 29 (2020), 2040003, 85 pages, arXiv:1701.06567.
  27. Gukov S., Putrov P., Vafa C., Fivebranes and 3-manifold homology, J. High Energy Phys. 2017 (2017), no. 7, 071, 81 pages, arXiv:1602.05302.
  28. Harper M., Verma modules over restricted quantum $\mathfrak{sl}_3$ at a fourth root of unity, arXiv:1911.00641.
  29. Harper M., sl3-root-of-1-invariant, https://github.com/mrhmath/sl3-root-of-1-invariant.
  30. Harper M., Kerler T., On Hopf ideals, integrality, and automorphisms of quantum groups at roots of 1, arXiv:2512.23503.
  31. Hedden M., Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277-2338, arXiv:math.GT/0606094.
  32. Horn P.D., On computing the first higher-order Alexander modules of knots, Exp. Math. 23 (2014), 153-169, arXiv:1303.1545.
  33. Ito T., A homological representation formula of colored Alexander invariants, Adv. Math. 289 (2016), 142-160, arXiv:1505.02841.
  34. Jiang B.J., On Conway's potential function for colored links, Acta Math. Sin. (Engl. Ser.) 32 (2016), 25-39, arXiv:1407.3081.
  35. Jones V.F.R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103-111.
  36. Kauffman L.H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417-471.
  37. Kauffman L.H., Saleur H., Free fermions and the Alexander-Conway polynomial, Comm. Math. Phys. 141 (1991), 293-327.
  38. Kohli B.-M., Patureau-Mirand B., Other quantum relatives of the Alexander polynomial through the Links-Gould invariants, Proc. Amer. Math. Soc. 145 (2017), 5419-5433, arXiv:1606.02538.
  39. Kohli B.-M., Tahar G., A lower bound for the genus of a knot using the Links-Gould invariant, arXiv:2310.15617.
  40. Kuperberg G., The quantum $G_2$ link invariant, Internat. J. Math. 5 (1994), 61-85, arXiv:math.GT/9201302.
  41. Lentner S., A Frobenius homomorphism for Lusztig's quantum groups for arbitrary roots of unity, Commun. Contemp. Math. 18 (2016), 1550040, 42, arXiv:1406.0865.
  42. Links J.R., Gould M.D., Two variable link polynomials from quantum supergroups, Lett. Math. Phys. 26 (1992), 187-198.
  43. Lopez Neumann D., van der Veen R., A plumbing-multiplicative function from the Links-Gould invariant, arXiv:2502.12899.
  44. Lusztig G., Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), 89-113.
  45. Martel J., Willetts S., Unified invariant of knots from homological braid action on Verma modules, Proc. Lond. Math. Soc. 128 (2024), e12599, 45 pages, arXiv:2112.15204.
  46. Morton H.R., Cromwell P.R., Distinguishing mutants by knot polynomials, J. Knot Theory Ramifications 5 (1996), 225-238.
  47. Murakami J., The multi-variable Alexander polynomial and a one-parameter family of representations of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{C}))$ at $q^2=-1$, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 350-353.
  48. Murakami J., A state model for the multivariable Alexander polynomial, Pacific J. Math. 157 (1993), 109-135.
  49. Ohtsuki T., Quantum invariants: A study of knots, 3-manifolds, and their sets, Ser. Knots Everything, Vol. 29, World Scientific Publishing, River Edge, NJ, 2002.
  50. Piccirillo L., The Conway knot is not slice, Ann. of Math. 191 (2020), 581-591, arXiv:1808.02923.
  51. Reshetikhin N.Yu., Turaev V.G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26.
  52. Rolfsen D., Knots and links, Math. Lecture Ser., Vol. 7, Publish or Perish, Inc., Berkeley, CA, 1976.
  53. Rupert M., Categories of weight modules for unrolled restricted quantum groups at roots of unity, Izv. Math. 86 (2022), 1204-1224, arXiv:1910.05922.
  54. Sartori A., The Alexander polynomial as quantum invariant of links, Ark. Mat. 53 (2015), 177-202, arXiv:1308.2047.
  55. The knot atlas, https://katlas.org.
  56. Turaev V., The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553.
  57. Turaev V., Quantum invariants of knots and 3-manifolds, De Gruyter Stud. Math., Vol. 18, de Gruyter, Berlin, 1994.
  58. Turaev V., Torsions of $3$-dimensional manifolds, Progress in Mathematics, Vol. 208, Birkhäuser, Basel, 2002.
  59. Viro O.Ya., Quantum relatives of the Alexander polynomial, St. Petersburg Math. J. 18 (2006), 391-457, arXiv:math.GT/0204290.
  60. Wada M., Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), 241-256.

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