On the AJ-conjecture for certain families of satellite knots
Ruppe, Dennis Aaron
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We investigate the AJ-conjecture for various families of satellite knots. We use explicit formulas for the colored Jones polynomials of cabled knots over torus knots to obtain a recurrence relation, then show that this recurrence is of minimal order for most of the knots in the family to verify the AJ-conjecture. We then examine cabled knots over the figure eight knot. Given a recurrence relation for the colored Jones polynomials of the figure eight knot, we provide a procedure to obtain such a relation for any of its cabled knots. We verify the AJ-conjecture by showing that this procedure when specialized at t = –1 also computes the A -polynomial of the knot. Finally, we turn our attention to Whitehead doubles of torus knots. The A -polynomials of these knots are computed and found to be given in terms of the A -polynomials of the twist knots. We then use the theory of holonomic functions to show that the recurrence relations for the colored Jones polynomials of these Whitehead doubles has an annihilator that, when evaluated at t = –1, is divisible by the A -polynomial of the knot.