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

SIGMA 19 (2023), 002, 12 pages      arXiv:2101.11708      https://doi.org/10.3842/SIGMA.2023.002

A Cable Knot and BPS-Series

John Chae
Department of Mathematics, Univeristy of California Davis, Davis, USA

Received August 03, 2022, in final form January 05, 2023; Published online January 13, 2023

Abstract
A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative simple method of finding the former series invariant.

Key words: knot complement; quantum invariant; $q$-series; Chern-Simons theory; categorification.

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References

  1. Bar-Natan D., Garoufalidis S., On the Melvin-Morton-Rozansky conjecture, Invent. Math. 125 (1996), 103-133.
  2. Chae J., Ancillary files for ''A cable knot and BPS series'', arXiv:2101.11708.
  3. Cooper D., Culler M., Gillet H., Long D.D., Shalen P.B., Plane curves associated to character varieties of $3$-manifolds, Invent. Math. 118 (1994), 47-84.
  4. Cooper D., Long D.D., Remarks on the $A$-polynomial of a knot, J. Knot Theory Ramifications 5 (1996), 609-628.
  5. Dimofte T., Gukov S., Lenells J., Zagier D., Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Number Theory Phys. 3 (2009), 363-443, arXiv:0903.2472.
  6. Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Stošić M., Sulkowski P., Branches, quivers, and ideals for knot complements, J. Geom. Phys. 177 (2022), 104520, 75 pages, arXiv:2110.13768.
  7. Garoufalidis S., On the characteristic and deformation varieties of a knot, in Proceedings of the Casson Fest, Geom. Topol. Monogr., Vol. 7, Geom. Topol. Publ., Coventry, 2004, 291-309, arXiv:math.GT/0306230.
  8. Garoufalidis S., Koutschan C., The noncommutative $A$-polynomial of $(-2,3,n)$ pretzel knots, Exp. Math. 21 (2012), 241-251, arXiv:1101.2844.
  9. Garoufalidis S., Lê T.T.Q., The colored Jones function is $q$-holonomic, Geom. Topol. 9 (2005), 1253-1293, arXiv:math.GT/0309214.
  10. Garoufalidis S., Sun X., The non-commutative $A$-polynomial of twist knots, J. Knot Theory Ramifications 19 (2010), 1571-1595, arXiv:0802.4074.
  11. Gukov S., Three-dimensional quantum gravity, Chern-Simons theory, and the $A$-polynomial, Comm. Math. Phys. 255 (2005), 577-627, arXiv:hep-th/0306165.
  12. Gukov S., Hsin P.S., Nakajima H., Park S., Pei D., Sopenko N., Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants, J. Geom. Phys. 168 (2021), 104311, 22 pages, arXiv:2005.05347.
  13. Gukov S., Manolescu C., A two-variable series for knot complements, Quantum Topol. 12 (2021), 1-109, arXiv:1904.06057.
  14. Gukov S., Park S., Putrov P., Cobordism invariants from BPS $q$-series, Ann. Henri Poincaré 22 (2021), 4173-4203, arXiv:2009.11874.
  15. 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.
  16. Gukov S., Putrov P., Vafa C., Fivebranes and 3-manifold homology, J. High Energy Phys. 2017 (2017), no. 7, 071, 80 pages, arXiv:1602.05302.
  17. Gukov S., Sulkowski P., A-polynomial, B-model, and quantization, J. High Energy Phys. 2012 (2012), no. 2, 070, 56 pages, arXiv:1108.0002.
  18. Hedden M., On knot Floer homology and cabling, Algebr. Geom. Topol. 5 (2005), 1197-1222, arXiv:math.GT/0406402.
  19. Hikami K., Difference equation of the colored Jones polynomial for torus knot, Internat. J. Math. 15 (2004), 959-965, arXiv:math.GT/0403224.
  20. Kucharski P., Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal N = 2$ theories, J. High Energy Phys. 2020 (2020), no. 9, 075, 26 pages, arXiv:2005.13394.
  21. Lê T.T.Q., Tran A.T., On the AJ conjecture for knots (with an appendix written jointly with Vu Q. Huynh), Indiana Univ. Math. J. 64 (2015), 1103-1151, arXiv:1111.5258.
  22. Levine A.S., Nonsurjective satellite operators and piecewise-linear concordance, Forum Math. Sigma 4 (2016), e34, 47 pages, arXiv:1405.1125.
  23. Melvin P.M., Morton H.R., The coloured Jones function, Comm. Math. Phys. 169 (1995), 501-520.
  24. Miller A.N., Homomorphism obstructions for satellite maps, arXiv:1910.03461.
  25. Miller A.N., Piccirillo L., Knot traces and concordance, J. Topol. 11 (2018), 201-220, arXiv:1702.03974.
  26. Park S., Inverted state sums, inverted Habiro series, and indefinite theta functions, arXiv:2106.03942.
  27. Park S., Large color $R$-matrix for knot complements and strange identities, J. Knot Theory Ramifications 29 (2020), 2050097, 32 pages, arXiv:2004.02087.
  28. Reshetikhin N.Yu., Turaev V.G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26.
  29. Reshetikhin N.Yu., Turaev V.G., Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
  30. Rozansky L., Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial, Comm. Math. Phys. 183 (1997), 291-306, arXiv:q-alg/9601009.
  31. Rozansky L., The universal $R$-matrix, Burau representation, and the Melvin-Morton expansion of the colored Jones polynomial, Adv. Math. 134 (1998), 1-31, arXiv:q-alg/9604005.
  32. Ruppe D., The AJ-conjecture and cabled knots over the figure eight knot, Topology Appl. 188 (2015), 27-50, arXiv:1405.3887.
  33. Ruppe D., Zhang X., The AJ-conjecture and cabled knots over torus knots, J. Knot Theory Ramifications 24 (2015), 1550051, 24 pages, arXiv:1403.1858.
  34. Tran A.T., On the AJ conjecture for cables of the figure eight knot, New York J. Math. 20 (2014), 727-741, arXiv:1405.4055.
  35. Tran A.T., On the AJ conjecture for cables of twist knots, Fund. Math. 230 (2015), 291-307, arXiv:1409.6071.
  36. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.

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