TITLE: Composite knots and their braid words
QUESTION [5 upvotes]: Given a composite knot K = K_1 # K_2, I wonder how the braid word looks like. Is it possible to see from the word that the knot is composite?
I am not aware of a statement such as "the closure of a braid of a given form is a composite knot" and wouldn't hope for it. But then there is Vogel's algorithm, perhaps something similar more advanced can give an organised way to keep the sum visible somehow when bringing the knot K into a braid presentation?

REPLY [5 votes]: Birman and Menasco show that the `composite braid representation' described by Marco Golla in his comment is in some sense canonical. That is, you can get from an arbitrary braid word to the one that Marco describes by a series of moves. See Birman-Menasco, Studying links via closed braids. IV. Composite links and split links, Invent. Math. 102 (1990), no. 1, 115–139.
There is also a later erratum in Invent. Math. 160 (2005), no. 2, 447–452 that corrects the proof; the statement remains the same. The erratum refers to a lot of subsequent work that might be useful.