Cosmetic surgery on the Figure Eight Sister
The double branched cover of the closure of the braid on the left is the lens space L(5,1). The braid axis lifts to the Figure Eight Sister. Adding in two full twists downstairs is equivalent to doing +1 surgery on the knot up stairs. After braid moves, we obtain the mirror.
Hence the figure eight sister exterior has two fillings yielding homeomorphic lens spaces.