Georgia Institute of Technology College of Sciences

The moduli space of representations of a fundamental group of a knot in SL(2,C) is an affine algebraic variety, and generically a complex curve, with an explicit projection to C^2. The ideal that defines this curve has special type described by binomial and linear equations. I will motivate this curve using elementary hyperbolic geometry, and its Newton polygon in the plane using geometric topology. Finally, I will describe a heuristic method for computing the Newton polygon without computing the curve itself, using tropical implitization, work in progress with Josephine Yu. The talk will be concrete, with examples of concrete curves that come from knots. This talk involves classical mathematics. A sequel of it will discuss quantum character varieties of knots and tropical curves.