On the tail of Jones polynomials of closed braids with a full twist

Champanerkar, Abhijit; Kofman, Ilya

doi:10.1090/S0002-9939-2013-11555-8

On the tail of Jones polynomials of closed braids with a full twist
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by Abhijit Champanerkar and Ilya Kofman

Proc. Amer. Math. Soc. 141 (2013), 2557-2567

DOI: https://doi.org/10.1090/S0002-9939-2013-11555-8

Published electronically: March 12, 2013

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Abstract:

For a closed $n$–braid $L$ with a full positive twist and with $\ell$ negative crossings, $0\leq \ell \leq n$, we determine the first $n-\ell +1$ terms of the Jones polynomial $V_L(t)$. We show that $V_L(t)$ satisfies a braid index constraint, which is a gap of length at least $n-\ell$ between the first two nonzero coefficients of $(1-t^2) V_L(t)$. For a closed positive $n$–braid with a full positive twist, we extend our results to the colored Jones polynomials. For $N>n-1$, we determine the first $n(N-1)+1$ terms of the normalized $N$–th colored Jones polynomial.

References

C. Armond. Walks along braids and the colored Jones polynomial, arXiv:1101.3810v1 [math.GT], 2011.
Joan Birman and Ilya Kofman, A new twist on Lorenz links, J. Topol. 2 (2009), no. 2, 227–248. MR 2529294, DOI 10.1112/jtopol/jtp007
Abhijit Champanerkar, David Futer, Ilya Kofman, Walter Neumann, and Jessica S. Purcell, Volume bounds for generalized twisted torus links, Math. Res. Lett. 18 (2011), no. 6, 1097–1120. MR 2915470, DOI 10.4310/MRL.2011.v18.n6.a5
Abhijit Champanerkar and Ilya Kofman, On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol. 5 (2005), 1–22. MR 2135542, DOI 10.2140/agt.2005.5.1
Abhijit Champanerkar, Ilya Kofman, and Eric Patterson, The next simplest hyperbolic knots, J. Knot Theory Ramifications 13 (2004), no. 7, 965–987. MR 2101238, DOI 10.1142/S021821650400355X
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, DOI 10.1112/S0010437X06002296
John Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), no. 1, 97–108. MR 896009, DOI 10.1090/S0002-9947-1987-0896009-2
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
David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Cusp areas of Farey manifolds and applications to knot theory, Int. Math. Res. Not. IMRN 23 (2010), 4434–4497. MR 2739802, DOI 10.1093/imrn/rnq037
Stavros Garoufalidis, The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011), no. 2, Paper 4, 23. MR 2795781, DOI 10.37236/2000
J.-M. Isidro, J. M. F. Labastida, and A. V. Ramallo, Polynomials for torus links from Chern-Simons gauge theories, Nuclear Phys. B 398 (1993), no. 1, 187–236. MR 1222806, DOI 10.1016/0550-3213(93)90632-Y
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
Robion Kirby and Paul Melvin, The $3$-manifold invariants of Witten and Reshetikhin-Turaev for $\textrm {sl}(2,\textbf {C})$, Invent. Math. 105 (1991), no. 3, 473–545. MR 1117149, DOI 10.1007/BF01232277
H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107–109. MR 809504, DOI 10.1017/S0305004100063982
A. Stoimenow, On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks, Trans. Amer. Math. Soc. 354 (2002), no. 10, 3927–3954. MR 1926860, DOI 10.1090/S0002-9947-02-03022-2
Hans Wenzl, Braids and invariants of $3$-manifolds, Invent. Math. 114 (1993), no. 2, 235–275. MR 1240638, DOI 10.1007/BF01232670