Mirrors and Ribbons

The connect sum of a knot and its mirror is the basic example of a slice knot.

Picture 1

Put one above the xy-plane and mirror it below.

Picture 5

The mirroring sweeps out an immersed annulus.

Picture 4

Note how the annulus runs through itself. These self-intersections are ribbon.

Picture 6

Snipping along a vertical arc of the annulus leaves an immersed disk. Its boundary is the connect sum of the knot and its mirror.

Picture 10 Picture 13
Picture 17 Picture 11

Picture 16

Picture 15

Some intersections may simply be slid away.
Picture 18
Others may not.
Picture 22

More images here.

~ by Ken Baker on April 20, 2009.

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