An alternative definition of the cell structure on the complement of a knot with diagram in
S2goes as follows.
Take the cell decomposition of S2 determined by the
quadrilateral decomposition and attached a
3-cell - thought of as the top hemisphere of S3.
Next attach an extra 2-cell (quadrilateral) for each crossing
by glueing a pair of parallel sides to the `coloured b` pair of parallel sides of the 2-cell in the
quadrilateral decomposition. Finally squeeze the other pair of parallel sides to a point
and squeeze all 0-cells to a common point so we end with a single 0-cell (and a single 3-cell) in the final complex.
See more on colouring
Alexander's approach was based on a different cell decomposition and may be thought of as an attempt to colour regions rather than colouring arcs.
Here is an extract from his paper. The rule is as shown. Notice only the direction on the underpass matters:
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