Mutual braiding and the band presentation of braid groups (Q2725555)

Suppose that \(A \cup B\) is a link consisting of two components \(A\) and \(B\). The link \(B\) is braided rel \(A\) when \(A\) is fibered and the fibration can be chosen so that \(B\) meets all fibers transversely. In addition, \(A \cup B\) is called a (generalized) exchangeable link if \(A\) and \(B\) are both fibered, \(B\) is braided rel \(A\) and \(A\) is braided rel \(B\). In case \(A\) is unknotted and the fibers form mutually open books [as defined by \textit{L. Rudolph}, Contemp. Math. 78, 657-673 (1988; Zbl 0669.57004)], \(A \cup B\) is called mutually braided. NEWLINENEWLINENEWLINEThis paper is concerned with detecting when a closed braid and its axis are mutually braided. It is shown that such a braid can be represented naturally as a word in the so-called band-generators of the braid group. This leads to a combinatorial method for deciding if a braid is mutually braided. This is applied to the problem of determining which Stallings braids are exchangeable.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].