The Jones slopes of a knot

Details The Jones slopes of a knot The Jones slopes of a knot

2009-11-18

arxiv.org/abs/0911.3627v6

Mathematics

Geometric Topology

19 pages, 2 figures

Scientific paper

The paper introduces 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, torus knots, the knots with at most 9 crossings, and the $(-2,3,n)$ pretzel knots.

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