Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Twist regions and coefficients stability of the colored Jones polynomial


Authors: Mohamed Elhamdadi, Mustafa Hajij and Masahico Saito
Journal: Trans. Amer. Math. Soc. 370 (2018), 5155-5177
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/tran/7128
Published electronically: February 8, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $ q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.


References [Enhancements On Off] (What's this?)

  • [1] Cody Armond, The head and tail conjecture for alternating knots, Algebr. Geom. Topol. 13 (2013), no. 5, 2809-2826. MR 3116304, https://doi.org/10.2140/agt.2013.13.2809
  • [2] Cody W. Armond and Oliver T. Dasbach, Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial, arXiv:1106.3948 (2011).
  • [3] Paul Beirne and Robert Osburn, $ q$-series and tails of colored Jones polynomials, Indag. Math. (N.S.) 28 (2017), no. 1, 247-260. MR 3597046, https://doi.org/10.1016/j.indag.2016.11.016
  • [4] Xuanting Cai and Robert G. Todd, A cellular basis for the generalized Temperley-Lieb algebra and Mahler measure, Topology Appl. 178 (2014), 107-124. MR 3276731, https://doi.org/10.1016/j.topol.2014.09.006
  • [5] Abhijit Champanerkar and Ilya Kofman, On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol. 5 (2005), 1-22. MR 2135542, https://doi.org/10.2140/agt.2005.5.1
  • [6] Abhijit Champanerkar and Ilya Kofman, On links with cyclotomic Jones polynomials, Algebr. Geom. Topol. 6 (2006), 1655-1668. MR 2253460, https://doi.org/10.2140/agt.2006.6.1655
  • [7] Oliver T. Dasbach and Xiao-Song Lin, On the head and the tail of the colored Jones polynomial, Compos. Math. 142 (2006), no. 5, 1332-1342. MR 2264669, https://doi.org/10.1112/S0010437X06002296
  • [8] Mohamed Elhamdadi and Mustafa Hajij, Pretzel Knots and $ q$-Series, Osaka J. Math. 54 (2017), no. 2, 363-381. MR 3657236
  • [9] David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008), no. 3, 429-464. MR 2396249
  • [10] David Futer, Efstratia Kalfagianni, and Jessica Purcell, Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics, vol. 2069, Springer, Heidelberg, 2013. MR 3024600
  • [11] Stavros Garoufalidis and Thang T. Q. Lê, Nahm sums, stability and the colored Jones polynomial, Res. Math. Sci. 2 (2015), Art. 1, 55. MR 3375651, https://doi.org/10.1186/2197-9847-2-1
  • [12] Mustafa Hajij, The tail of a quantum spin network, Ramanujan J. 40 (2016), no. 1, 135-176. MR 3485997, https://doi.org/10.1007/s11139-015-9705-9
  • [13] Mustafa Hajij, The Bubble skein element and applications, J. Knot Theory Ramifications 23 (2014), no. 14, 1450076, 30. MR 3312619, https://doi.org/10.1142/S021821651450076X
  • [14] Mustafa Hajij, The colored Kauffman skein relation and the head and tail of the colored Jones polynomial, J. Knot Theory Ramifications 26 (2017), no. 3, 1741002, 14. MR 3627702, https://doi.org/10.1142/S0218216517410024
  • [15] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1-25. MR 696688, https://doi.org/10.1007/BF01389127
  • [16] Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407. MR 899057, https://doi.org/10.1016/0040-9383(87)90009-7
  • [17] Adam Keilthy and Robert Osburn, Rogers-Ramanujan type identities for alternating knots, J. Number Theory 161 (2016), 255-280. MR 3435728, https://doi.org/10.1016/j.jnt.2015.02.002
  • [18] Marc Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. (3) 88 (2004), no. 1, 204-224. With an appendix by Ian Agol and Dylan Thurston. MR 2018964, https://doi.org/10.1112/S0024611503014291
  • [19] Ryan A. Landvoy, The Jones polynomial of pretzel knots and links, Topology Appl. 83 (1998), no. 2, 135-147. MR 1606488, https://doi.org/10.1016/S0166-8641(97)00100-4
  • [20] Christine Lee, Stability properties of the colored Jones polynomial, arXiv:1409.4457.
  • [21] Christine Lee, A trivial tail homology for non A-adequate links, arXiv:1611.00686v1.
  • [22] W. B. R. Lickorish, Calculations with the Temperley-Lieb algebra, Comment. Math. Helv. 67 (1992), no. 4, 571-591. MR 1185809, https://doi.org/10.1007/BF02566519
  • [23] G. Manchón and P. Maria, On the Kauffman bracket of pretzel links, Marie Curie Fellowships Annals 2 (2003), 118-122.
  • [24] G. Masbaum and P. Vogel, $ 3$-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994), no. 2, 361-381. MR 1272656
  • [25] Tomotada Ohtsuki, Quantum invariants, Series on Knots and Everything, vol. 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. A study of knots, 3-manifolds, and their sets. MR 1881401
  • [26] Józef H. Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999), no. 1, 45-66. MR 1723531
  • [27] N. Reshetikhin and V. G. Turaev, Invariants of $ 3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597. MR 1091619, https://doi.org/10.1007/BF01239527
  • [28] Lev Rozansky, Khovanov homology of a unicolored B-adequate link has a tail, Quantum Topol. 5 (2014), no. 4, 541-579. MR 3317343, https://doi.org/10.4171/QT/58
  • [29] Morwen B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93 (1988), no. 2, 285-296. MR 948102, https://doi.org/10.1007/BF01394334
  • [30] V. G. Turaev and O. Ya. Viro, State sum invariants of $ 3$-manifolds and quantum $ 6j$-symbols, Topology 31 (1992), no. 4, 865-902. MR 1191386, https://doi.org/10.1016/0040-9383(92)90015-A
  • [31] Katie Walsh, Higher Order Stability in the Coefficients of the Colored Jones Polynomial, arXiv preprint arXiv:1603.06957 (2016).
  • [32] Hans Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5-9. MR 873400
  • [33] Wolfram Research Inc., Mathematica, Wolfram Research, Inc. 10.4 (2016).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M27

Retrieve articles in all journals with MSC (2010): 57M27


Additional Information

Mohamed Elhamdadi
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33647
Email: emohamed@mail.usf.edu

Mustafa Hajij
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33647
Address at time of publication: Department of Computer Science and Engineering, University of South Florida, Tampa, Florida 33647
Email: mhajij@usf.edu

Masahico Saito
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33647
Email: saito@usf.edu

DOI: https://doi.org/10.1090/tran/7128
Received by editor(s): August 3, 2016
Received by editor(s) in revised form: November 14, 2016
Published electronically: February 8, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society