New York Journal of Mathematics
Volume 21 (2015) 905-941


Efstratia Kalfagianni and Anh T. Tran

Knot cabling and the degree of the colored Jones polynomial

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Published: September 16, 2015
Keywords: Adequate knot, boundary slope, cable knot, colored Jones polynomial, essential surface, Jones slope, Slope Conjecture, Strong Slope Conjecture
Subject: Primary 57N10. Secondary 57M25.

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.


E. K. was partially supported by NSF grants DMS-1105843 and DMS-1404754

Author information

Efstratia Kalfagianni:
Department of Mathematics, Michigan State University, East Lansing, MI, 48824

Anh T. Tran:
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080