6
$\begingroup$

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 and HERE.

  • Kauffman's extended bracked polynomial HERE.

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ Can you describe disoriented tangles by some universal property? $\endgroup$
    – Qiaochu Yuan
    Commented Jan 28, 2015 at 10:41
  • 2
    $\begingroup$ @Qiaochu: The free braided category with duals (spherical) generated by one anti-symmetrically self-dual object. (Probably my braided category adjectives are not quite right.) $\endgroup$
    – Kevin Walker
    Commented Jan 28, 2015 at 16:15

1 Answer 1

Reset to default
3
$\begingroup$

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. I strongly suspect there are earlier examples in papers of Reshetikhin and/or Kirillov and/or Turaev.

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .