There are several invariants whose "natural" domain is a category of **disoriented** tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the orientation is reversed.


*For example*:


* Khovanov homology [HERE](http://arxiv.org/pdf/math/0701339v2.pdf) and [HERE](http://arxiv.org/pdf/0707.3051.pdf).
* Kauffman's extended bracked polynomial [HERE](http://arxiv.org/pdf/0712.2546.pdf).


I was unable, however, to track down the origin of the idea- where were disorientations first used, and what is the correct citation to use to reference it outside the context of one of these invariants?


Further, I would like to ask whether there are other known contexts in which disoriented tangles of some flavour appear, other than the contexts mentioned above.



What is the original reference for disorientations on tangle diagrams?
The *term* "disoriented tangle" first appeared in arxiv:0701339v2 (your first link). That is also the earliest reference I know for bordisms of disoriented tangles.


But the *idea* of disoriented tangles is much older. It emerges naturally when one considers string diagrams for $Rep(U\_q sl\_2)$. See, for example, Figures 3.22 and 4.8 of Kirby and Melvin's paper [here](https://tqft.net/svn/d4/trunk/links-article/papers/kirby-melvin.pdf). I strongly suspect there are earlier examples in papers of Reshetikhin and/or Kirillov and/or Turaev.


