Figure Eight and Braid

The figure eight knot admits a free involution with the unknot as the fixed set. The once-punctured torus fiber is invariant under this involution too, though the unknot intersects it three times. Quotienting out by this involution we see the unknot becomes a 3-string braid, the figure eight knot becomes the braid axis, and the fiber becomes a disk bounded by this braid axis.

FE1

FE2 FE3
FE4 FE5
FE6 FE7
FE8 FE9
FE10 FE11
FE12 FE13

~ by Ken Baker on November 17, 2007.

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