The Jones Slopes of a Knot
نویسنده
چکیده
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.
منابع مشابه
Slopes and Colored Jones Polynomials of Adequate Knots
Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.
متن کاملSlopes and Colored Jones Polynomials of Adequate Links
Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. More precisely, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.
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متن کاملA Jones Slopes Characterization of Adequate Knots
We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of “Jones slopes” of knots and the essential surfaces that realize the slopes. For alternating knots the reformulated characterization follows by recent work of J. Greene and J. Howie. 20...
متن کاملKnot Cabling and the Degree of the Colored Jones Polynomial Ii
We continue our study of the degree of the colored Jones polynomial under knot cabling started in [8]. Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we verify Garoufalidis’ Slope Conjecture and a conjecture of [8] for cables of a two-parameter family of closed 3-braids called 2-fusion knots. 2010 Mathematic...
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تاریخ انتشار 2010