##
**The braid index of generalized cables.**
*(English)*
Zbl 0811.57013

Summary: 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 bridge number, though this was proved by H. Schubert [Math. Z. 61, 245-288 (1954; Zbl 0058.174)]. It was only with the advent of the Jones polynomial and its relatives in the mid 1980’s, 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.

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 bridge number, though this was proved by H. Schubert [Math. Z. 61, 245-288 (1954; Zbl 0058.174)]. It was only with the advent of the Jones polynomial and its relatives in the mid 1980’s, 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.

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |