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Stavros Garoufalidis and Roland van der Veen
Quadratic integer programming and the Slope Conjecture view print
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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. |
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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 of one sector.
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| Author information
Stavros Garoufalidis:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
stavros@math.gatech.edu
Roland van der Veen:
Mathematisch Intstituut, Leiden University, Leiden, Niels Bohrweg 1, The Netherlands
r.i.van.der.veen@math.leidenuniv.nl
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