Quantum Topology

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Volume 2, Issue 1, 2011, pp. 43–69
DOI: 10.4171/QT/13

The Jones slopes of a knot

Stavros Garoufalidis[1]

(1) School of Mathematics, Georgia Institute of Technology, GA 30332-0160, ATLANTA, UNITED STATES

The paper introduces the slope conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2,3,n) pretzel knots.

Keywords: Knot, link, Jones polynomial, Jones slope, Jones period, quasi-polynomial, alternating knots, signature, pretzel knots, polytopes, Newton polygon, incompressible surfaces, slope, slope conjecture

Garoufalidis Stavros: The Jones slopes of a knot. Quantum Topol. 2 (2011), 43-69. doi: 10.4171/QT/13