Cosmetic surgery on the Figure Eight Sister

Cosmetic surgery on the Figure Eight Sister, originally uploaded by epsilon_is_afraid_of_zeta.

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.

~ by Ken Baker on November 17, 2007.

2 Responses to “Cosmetic surgery on the Figure Eight Sister”

  1. Doesn’t this follow from the fact that the figure eight sibling is amphicheiral? I suppose the only Dehn filling which might not have a corresponding mirror Dehn filling is the slope fixed by the orientation reversing involution.

  2. Of course. This is but a method of explicitly observing the cosmetic surgery. One of the things I like about viewing manifolds as double branched covers of links is the ability to tangibly (or dare I say tanglebly) work with Dehn surgery.

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