The Jones slopes of a knot

By Stavros Garoufalidis. 2009

74 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/09fff3edbb935dc3ef35f41781c58a281829bcae

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Citations in the corpus

• 3-Manifolds and 3d Indices (2011, background): “ the operators L^ M are polynomials in ^x ;p^ ;and q, along with the assumption that indices I M(m;e;q) are q-series bounded from below, implies that the generic growth of lead(m;e) is quadratic (cf. [64]). This is essentially because the operators x^ 2;p^ only involve linear factors q m and qe 2 , ”