Quadratic integer programming and the slope conjecture

Abstract

The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a corresponding quadratic integer programming problem. We illustrate this principle for a 2-parameter family of 2-fusion knots. Combined with the results of Dunfield and the first author, this confirms the Slope Conjecture for the 2-fusion knots.

22 pages, 32 figures, 3 tables

Cited by in corpus (7)

• The slope conjecture for graph knots
• Slopes for Pretzel Knots
• Computing tropical curves via homotopy continuation
• Singularization of knots and closed braids
• Jones slopes and coarse volume of near-alternating links
• The degree of the colored HOMFLY polynomial
• Knot Cabling and the Degree of the Colored Jones Polynomial