Essentially, the first picture above is half of this picture. Interior branch point, order 2, index 1. When the branch point is on the boundary, only half of this surface is seen. The order of a boundary branch point must be even if the surface is to touch the boundary monotonically--with odd order, it would double back on itself.
Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold
Thus order 2 is the simplest case. Index 0 would give a piece of a plane, so index 1 is the simplest three-dimensional case. Thus order 2, index 1 goes around as an interior branch point 3 times while going up and down 4 times; and as a boundary branch point, it goes around one and a half times while going up and down twice.
An order 4, index 2 boundary branch point goes around two and a half times while going up and down three and a half times. These are the cases illustrated at the links above. The surfaces illustrated have simple formulas using a complex variable z.
A Theory of Branched Minimal Surfaces | Mathematical Association of America
Namely, if the surface is given by three functions X z , Y z , and Z z , then we have. Here Re means "the real part of". These formulas corroborate the English descriptions above using the phrase "goes around" so many times.
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Here is a good problem for the beginner at this point: How many lines of intersection of the surface with itself will there be, in a branch point of order M and index k? Any branch point must be given by formulas that start out like those above, but there may also be terms with higher powers of z. You will have noticed that if you tried to solve the exercise above.
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If there is to be any separation of the different "sheets" of the surface, that must arise from higher-order terms in the formula for the surface. Notice that at first glance the surface appears to be like a distorted disk--but looking carefully, you can see from the coordinate markings on the surface that it actually does go around one and half times, with the last half time exactly overlapping the first half.
Order 2, Index 3 A false branch point. If the overlap is exact, as in this picture, the branch point is called a false branch point.
But bear in mind that the picture could look almost like this, with the sheets being separated by a very tiny amount that might not even be visible in a computer-graphic picture due to a term in a high power of z. A branch point that is not false is called, naturally, a true branch point.
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