Thursday, February 03, 2005

Unknotting...

With this I hope to start a series of posts on papers/ideas that I found interesting at SODA 2005. One of the side effects of no wireless access at SODA was that I couldn't comment on talks as they happened, but it (hopefully) meant that I had time to reflect on what I found interesting.

Exhibit 1, in no particular order, is the following result, by Hara, Tani and Yamamoto:

Unknotting is in AM co-AM

Unknotting is one of the more fundamental problems in knot theory. If we think of a (tame) knot as a (really tangled) loop in three dimensions, then much of simple knot theory focuses on the question: Given two knots, are they equivalent ? (where equivalence is expressed in terms of homeomorphisms).

The trivial knot (or the "unknot") is the simple circle in 3D, and thus the unknotting problem is:

UNKNOTTING: Given a knot, is it equivalent to the unknot ?

It was first shown in 1961 by Haken that Unknotting was decidable. After a brief gap of 36 years, Haas, Lagarias and Pippenger showed that Unknotting was in fact in NP (!), and conjectured that it was in NP ∩ co-NP. Other results along the way showed that Genus (is the genus of a knot less than g) was in PSPACE, and ManifoldGenus (a generalization to a given arbitrary manifold), was NP-Complete.

The new result speaks for itself. Formally, the authors show that Knotting (the complement of Unknotting) is in AM, by presenting a two-round interactive protocol for it. They also mention a corollary that if Unknotting is NP-Complete, then PH collapses to the second level. If I had paid better attention in my complexity class I would have figured out how this follows, but alas... I'd be grateful for any enlightenment. (Update: here it is)

The tragedy for me is that although it is clear that this result is interesting, the proof itself is rather impenetrable, requiring a knowledge of combinatorial topology above and beyond what I can muster up. This is why translators are needed ! (Update: here it is)

Knot theory is a good example of a (relatively arcane) subfield of mathematics that has suddenly become profoundly important in "real life". For more on this, read what well-known theoretical computer scientist John Baez has to say :).

On a more frivolous note, you can do knot art ! Calling our resident mathematical knot knitter...