1.2. KNOTTED SURFACE DIAGRAMS

3

FIGURE

2. Projections and broken diagrams of knotted surfaces

defining this curve is to use configuration spaces. Let

Cr = {(xi,..., xr) : Xj £ F & Xj ^ Xk for j ^ k

&f(xi) = f(x2) = --- = f(xr)}.

This is a space on which there is a free action of the permutation group E

r

. The

associated r-fold cover Cr = Cr x^

r

{1,2, ...,r } is called the r-decker manifold

(double decker or triple decker manifold when r = 2 or 3). The quotient T(r) =

Cr/Yir is the r-tuple manifold. Evidently, there are maps

fr:Cr—^F

and

fr : T(r) - R

3

defined by

fr([{xi,...,Xr)]j]) =Xj

and

M[xiy-,Xr]) = /(^l)