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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Abhijit Champanerkar and Ilya Kofman
Journal: Proc. Amer. Math. Soc. 141 (2013), 2557-2567
MSC (2010): Primary 57M25
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.


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Additional Information

Abhijit Champanerkar
Affiliation: Department of Mathematics, College of Staten Island, City University of New York, Staten Island, New York 10314 – and – Department of Mathematics, Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: abhijit@math.csi.cuny.edu

Ilya Kofman
Affiliation: Department of Mathematics, College of Staten Island, City University of New York, Staten Island, New York 10314 – and – Department of Mathematics, Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: ikofman@math.csi.cuny.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11555-8
PII: S 0002-9939(2013)11555-8
Received by editor(s): April 5, 2011
Received by editor(s) in revised form: October 15, 2011
Published electronically: March 12, 2013
Additional Notes: Both authors gratefully acknowledge support by the NSF, Simons Foundation, and PSC-CUNY
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.