From hyperbolic volume computation, I found that the following two 3-manifolds are (possibly orientation-reversal) homeomorphic:

  1. surgery on figure-eight knot $4_1$, with slope $-5$, and
  2. surgery on $5_2$ knot with slope $5$.

Is there simple way to show it using Kirby calculus? It must be easy but I am not that familiar with Kirby calculus.


Yes, there is a simple way. Below is a sequence of pictures illustrating the procedure (created using Kirby calculator).

$5_2$: This is $5_2$ Blowup at the clasp: This is the blowup Isotopy: After the isotopy Blowdown the purple unknot: And this is after blowing down the purple curve

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  • $\begingroup$ I couldn't follow the isotopy passage. Is it possible to explain it? $\endgroup$ – user150450 Feb 1 at 23:07
  • 1
    $\begingroup$ I'm just claiming that the two links (above and below "Isotopy:") differ by a link isotopy in $S^3$. Just undo the kinks of the purple component (i.e. do Reidemeister 1 moves) dragging the golden component along. $\endgroup$ – Marco Golla Feb 1 at 23:55

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