Given a link L in Â3
there is an orientable surface
S with one boundary curve embedded in Â3
in such a way that the boundary of S is the link L. Take a projection L'
of L, orient it by selecting directions of curves indicated by arrows.
Then, at each crossing, add two extra directed arcs (x1,x2)
and (y1,y2) bypassing the crossing, but compatible
with the orientation (please compare this operation with the derivation
of mirror curves, page 232) (the next figure, a). Then delete all crossings,
leaving a set of oriented circles called Seifert circles (b).
Seifert circles obtained may or may not be nested. Two Seifert circles
are nested if one of them is inside the other and if orientations
of the two circles coincide. When Siefert circles obtained are not nested
(b), the change-of-infinity operation- placing of the
inside of the selected Seifert circle, nests the two circles (c).
The genus of the surface S constructed from L that way is g = [(v-s+2)/2],
where v is the number of crossings in L', and s is the number of Seifert
circles arising from L. Different projections of L might give different
surfaces, and hence a different g. The genus of a link L is the
minimal genus of all orientable surfaces which L spans (Gilbert and Porter, 1994;
Sossinsky, 2002). For a direct product the equality g(L1#L2)
= g(L1)+g(L2) holds.