Knot Cabling and the Degree of the Colored Jones Polynomial

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Knot Cabling and the Degree of the Colored Jones Polynomial

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Title: Knot Cabling and the Degree of the Colored Jones Polynomial
Author(s):
Kalfagianni, E.;
Tran, Anh T. (UT Dallas)
Item Type: Article
Keywords: Adequate knots
Satellite knots
Cable knots
Jones polynomials
Abstract: We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot K satisfies the Slope Conjecture then a (p, q)-cable of K satisfies the conjecture, provided that p/q is not a Jones slope of K. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture 5.1), and we prove it for the following classes of knots: iterated torus knots and iterated cables of adequate knots, iterated cables of several nonalternating knots with up to nine crossings, pretzel knots of type (-2, 3, p) and their cables, and two-fusion knots.
Publisher: University at Albany
ISSN: 1076-9803
Persistent Link: http://hdl.handle.net/10735.1/4920
http://nyjm.albany.edu/j/2015/21-41v.pdf
Bibliographic Citation: Kalfagianni, E., and A. T. Tran. 2015. "Knot cabling and the degree of the colored Jones polynomial." New York Journal of Mathematics 21, retrieved from http://nyjm.albany.edu/j/2015/21-41.html
Terms of Use: ©2015 University of Albany. All rights reseerved.
Sponsors: E. K. was partially supported in part by NSF grants DMS{1105843 and DMS{1404754.

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