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The slope conjecture for graph knots



The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the coloured Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.



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[1] Dunfield, N.M. and Garoufalidis, S. Incompressibility criteria for spun-normal surfaces. Trans. Amer. Math. Soc. 364 (2012), 61096137.
[2] Futer, D., Kalfagianni, E. and Purcell, J. Slopes and colored Jones polynomials of adequate knots. Proc. Amer. Math. Soc. 139 (2011), 18891896.
[3] Garoufalidis, S. The Jones slopes of a knot. Quantum Topology 2 (2011), 4369.
[4] Garoufalidis, S. and Le, T.T. The colored Jones function is q-holonomic. Geom. Topol. 9 (2005), 12531293.
[5] Garoufalidis, S. and van der Veen, R. Quadratic integer programming and the slope conjecture. arXiv:1405.5088.
[6] Gordon, C.McA. Dehn surgery and satellite knots. Trans. Amer. Math. Soc. 275 (1983), 687708.
[7] Gromov, M. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1983), 213307.
[8] Hatcher, A.E. On the boundary curves of incompressible surfaces. Pacific J. Math. 99 (1982), 373377.
[9] Hatcher, A.E. Notes on basic 3–manifold topology. (2000) Freely available at
[10] Kalfagianni, E. and Tran, A.T. Knot cabling and the degree of the coloured Jones polynomial. New York J. Math. 21 (2015), 905941.
[11] Klaff, B. and Shalen, P.B. The diameter of the set of boundary slopes of a knot. Algebr. Geom. Topol. 6 (2006), 10951112.
[12] Lee, C. and van der Veen, R. Slopes for pretzel knots. arXiv:1602.04546.
[13] Morton, H. The coloured Jones function and Alexander polynomial for torus knots. Math. Proc. Camb. Phil. Soc. 117 (1995), 129135.
[14] Soma, T. The Gromov invariant of links. Invent. Math. 64 (1981), 445454.
[15] Thurston, W.P. The geometry and topology of 3-manifolds. Lecture notes (Princeton University, 1979).
[16] van der Veen, R. A cabling formula for the colored Jones polynomial. arXiv:0807.2679.


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