A sequence of rational functions in a variable q is q-holonomic (see [J. Comput. Appl. Math. 32, No.3, 321-368 (1990; Zbl 0738.33001)] and [Invent. Math. 103, No.3, 575-634 (1991; Zbl 0739.05007)]) if it satisfies a linear recursion with coefficient polynomials in q and qn. In virtue of a fundamental result by Wilf-Zeilberger, Quantum Topology turns out to provide us with a plethora of q-holonomic sequences of natural origin. In particular, the present paper takes into account the q-holonomic sequence of Jones polynomials of a knot and its parallels (see [Geom. Topol. 9, 1253-1293 (2005; Zbl 1078.57012)]). The author associates a tropical curve (see [Contemporary Mathematics 377, 289-317 (2005; Zbl 1093.14080)] and [Math. Mag. 82, No. 3, 163-173 (2009; Zbl 1227.14051)]) to each q-holonomic sequence; in particular, to every knot K a tropical curve is associated, via the Jones polynomial of K and its parallels. As a consequence, a relation is established between the AJ Conjecture ([Geometry and Topology Monographs 7, 291-309 (2004; Zbl 1080.57014)]) and the Slope Conjecture ([Quantum Topol. 2, No. 1, 43-69 (2011; Zbl 1228.57004)]), which relate the Jones polynomial of K and its parallels respectively to the SL(2;C) character variety and to slopes of incompressible surfaces. The paper gives also an explicit computation of the tropical curve for the 41, 52 and 61 knots, verifying in these cases the duality between the tropical curve and a Newton subdivision of the A-polynomial of the knot.
|Data di pubblicazione:||2012|
|Titolo:||REVIEW OF: "Garoufalidis Stavros, Knots and tropical curves, Champagnerkar, Abhijit (ed.) et al., Interactions between hyperbolic geometry, quantum topology and number theory. Proceedings of a workshop, June 3–13, 2009 and a conference, June 15–19, 2009, Columbia University, New York, NY, USA. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218- 4960-6/pbk). Contemporary Mathematics 541, 83-101 (2011)." [DE059544674]|
|Autori:||Casali, Maria Rita|
|Autori del volume:||Garoufalidis, Stavros|
|Appare nelle tipologie:||Recensione in Rivista|
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