categories: Re: Tangle, Braid... related category?

[Duraid Madina <duraid@fl.net.au>]

> There are various ways this category could be formulated (are the
> strings allowed to cross each other? are they allowed to double back?
> etc), but my question is: has anything been written about it?  Does it 
> have a name? Does it remind anyone of another category which has been
> studied?

I vaguely recall these being referred to as "vines", and I think a text by
Joan Birman dealt with them a little. I'm not sure as to how far these
things were "categorified", though.
 
Sorry for being so vague,
 
        Duraid