  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.
