Knots and tropical curves

Details Knots and tropical curves Knots and tropical curves

2010-03-23

arxiv.org/abs/1003.4436v3

Mathematics

Geometric Topology

16 pages, 7 figures

Scientific paper

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial of the knot. We compute explicitly the tropical curve for the $4_1$, $5_2$ and $6_1$ knots and verify the above prediction.

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