The square and granny tangles.

Sometime after tying your shoelaces and an overhand knot in a string you might have learned about the square knot:

And you might have tied the granny knot instead:

Well, if one were physically tying ropes together the ends might more naturally have gone out vertically up and down rather than off to the sides. The square knot is used to hold two ropes together. The granny knot is when you do it wrong… it’s more prone to slip. The funny thing is that the square knot can be closed up to form a split link whereas the granny knot can’t. Okay, so this is comparing physical versus topological properties.

Let’s look a bit more at the topology.

We have a particular rational tangle: three vertical twists in a ball. It’s a rational tangle because there’s a disk that separates the two unknotted strands.

Dang. WordPress doesn’t allow iframes or javascript.
Rotate it here.

Chopping along this disk gives two balls, each with a trivial arc.

Rotate.

Joining this tangle to its mirror (the tangle sum) gives the square tangle.

The disks glue up to form a thrice-punctured sphere.

Spin!

Joining two copies of this tangle together (the tangle sum) gives the granny tangle.

The disks glue up to form a once-punctured torus.

Give it a whirl.

If the granny tangle were to be found in a two component split link, then its strands must belong to just one component. Otherwise the splitting sphere would intersect the ball of the granny tangle in a planar surface separating the two arcs, but any such surface necessarily should have genus as suggested above.

~ by Ken Baker on February 24, 2009.