The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels with one of the 2-handles. So we are left with two 1-handles and a 2-handle. I want to how the handle diagram of this knot complement looks like, more specifically I cannot figure out how the attaching circle of the 2-handle is linked with the 1-handles.
2026-03-26 23:01:07.1774566067
figure-8 knot complement
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The figure 8 has knot group $<x,y| yxy^{-1}xyx^{-1}y^{-1}xy^{-1}x^{-1}>$. Using the fact that the Kirby diagram is going to give us a presentation of the fundamental group of the complement (aka the knot group) we can draw the diagram for the complement. The two one handles will correspond to the generators and the two handle will give us that relation. Here's a quick sketch of the diagram.