Knots with infinitely many non-characterizing slopes

Using the techniques on annulus twists, we observe that 6₃ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots 6₂, 6₃, 7₆, 7₇, 8₁, 8₃, 8₄, 8₆, 8₇, 8₉, 8₁₀, 8₁₁, 8₁₂, 8₁₃, 8₁₄, 8₁₇, 8₂₀ and 8₂₁ have infinitely many non-characterizing slopes. We also introduce the notion of trivial annulus twists and give some possible applications. Finally, we completely determine which knots have special annulus presentations up to 8-crossings.