Monday, August 22, 2011

All baskets are closed

Baskets lie at the transition between knots and surfaces. Put another way, it is in contemplating baskets that the questions we ask about knots begin to supplant the questions we ask about surfaces. As a case in point, it can seem arbitrary to insist (as is always done in map theory) that a surface be closed, i.e., that it can have no boundary. On the other hand, in knot theory it would be absurd to posit a knot with a boundary.

It may seem intuitive that baskets need not be closed (what earthly use would a closed basket be?) But, in truth all baskets are closed. To see this we have to look more closely at the nature of the selvage, the proper weaving termination of a basket at its mouth.

Here are representations of properly selvaged basket openings of 3, 4, and 6 sides.

They can be tiled together to form a plain (i.e., strictly over-and-under) weaving:

Every plain-woven basket is entirely composed of such properly selvaged openings. There is no other sort of opening or boundary in a properly selvaged basket. In that sense, all baskets are boundary-less and therefore closed.

No comments: