The Minkowski units of ribbon knots

Abstract

2. Ribbon projections. Let k be a ribbon knot in R3, let 12 be the unit square, and let f: 12 -R3 be a ribbon whose boundary is k. By definition of a ribbon, the singular set S= {xflff(x) xx} of f consists of an even number 2n of pairwise disjoint arcs. Half of these arcs, say C1, * * , C., have the property that both their endpoints lie on 2, but they are otherwise disjoint from J2. For each Ci, i= 1, * , n there is a unique arc C in S such that f(Ci) =f(Ci) and C! fI2 The arcs C* separate f2 into n +1 components Xo, * * , Xn, and the endpoints of the arcs Ci separate 12 into a number of component arcs. At least two of these component arcs, say E1, * * *, Em, have the property that they join the two endpoints of some arc Ci. For each i=1, = * , n, let pi be an interior point of Ci and let p be the unique point of C' such that f(p) =f(pi). Also let qi be an interior point of Ei for each i= 1, * , m. Then each component Xi of I2-Ufo1 Ci has a certain number r of pi's and qi's designated on its boundary and a certain number s of p!'s designated on its interior.