The Strong Slope Conjecture and crossing numbers for Mazur doubles of knots

TL;DR

This paper proves that the Strong Slope Conjecture holds for Mazur doubles of knots under certain conditions, and explores implications for crossing numbers and potential counterexamples related to Jones slopes.

Contribution

It establishes the Strong Slope Conjecture for Mazur doubles of knots and derives crossing number bounds for certain doubled knots, extending previous conjectures.

Findings

Mazur doubles satisfy the Strong Slope Conjecture if the original knot does.

Crossing number of Mazur doubles of adequate knots with trivial writhe is either 9 times the original crossing number plus 2 or 3.

Potential counterexamples to the Strong Slope Conjecture are linked to knots with Jones slopes less than -1/4.

Abstract

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that Mazur doubles of knots satisfy the Strong Slope Conjecture if the original knot does. Consequently, any knot obtained by a finite sequence of cabling, untwisted w--generalized Whitehead doublings with w > 0, connected sums and Mazur doublings of B--adequate knots or torus knots satisfies the Strong Slope Conjecture. On the other hand, it may be worth mentioning that under these hypotheses, if there exists a knot with a Jones slope less than -1/4, then its Mazur double would either provide a counterexample to the Strong Slope Conjecture or have a Jones surface that is unrelated to any Jones surface of the knot. Following work of Kalfagianni and Lee, we also use our results to show that the Mazur double of an adequate knot K with trivial writhe has crossing number either 9c(K)+2 or 9c(K)+3.