If one knot is fashioned into
another, by replacing each strand with q strands, then something gets multiplied by
q. What? The answer should not be overly dependent on how these strands are
intertwined. We show that an invariant called the braid index is an answer. This
proposition is apparently new. Another answer covered by our proof is the bridgenumber, though this was proved by Shubert in 1954. It was only with the
advent of the Jones polynomial and its relatives in the mid 1980s, that much
attention has been given to the braid index. For example, the knots obtained by
repeated period doubling were shown to obey the multiplication rule, though no
one seems to have thought of it this way. Their braid indices are powers
of 2. We first considered the current proposition in trying to show that a
certain knot, known to have braid index 5, could not be a two-cabling of
anything.
