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)
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