New York Journal of Mathematics
Volume 22 (2016) 907-932


Stavros Garoufalidis and Roland van der Veen

Quadratic integer programming and the Slope Conjecture

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Published: September 9, 2016
Keywords: knot, link, Jones polynomial, Jones slope, quasi-polynomial, pretzel knots, fusion, fusion number of a knot, polytopes, incompressible surfaces, slope, tropicalization, state sums, tight state sums, almost tight state sums, regular ideal octahedron, quadratic integer programming.
Subject: Primary 57N10. Secondary 57M25.

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 of one sector.

Author information

Stavros Garoufalidis:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

Roland van der Veen:
Mathematisch Intstituut, Leiden University, Leiden, Niels Bohrweg 1, The Netherlands