Dylan: A Question by Mrowka on Pseudo-Diagrams
June-16-11
12:27 PM
Date: Wed, 15 Jun 2011 14:44:51 -0400
From: Dylan Thurston <...>
To: Dror Bar-Natan <drorbn@math.toronto.edu>
Subject: Pseudo-diagrams
Hello Dror,
In a talk of Tom Mrowka, I just heard a knot theory question that seems up our alley.
Definition.  A _pseudo-diagram_ for K in S^3 is an embedded 4-valent graph G in Y with a distinguished choice of 3 resolutions 0, +, infty for each 4-valent vertex, so that
  (a) The three resolutions look locally like the standard 3 resolutions;
  (b) If you resolve G with all + resolutions, you get K;
  (c) If you resolve G with all 0 or infty resolutions (in any combination), you get an unlink.
Two pseudo-diagrams D, D' are related by _forgetting_ a crossing if D' is obtained from D by doing a + resolution at one crossing.
Question: Are any two pseudo-diagrams for K related by a sequence of (un)forgetting operations?
Every standard diagram gives a pseudo-diagram in an obvious way, and the resulting pseudo-diagrams are related by (un)forgetting operations.  (Tom claimed that, in a standard diagram, you can always forget two of the three crossings near a Reidemeister III move.  You can then do an isotopy of the graph and unforget two new crossings to get to the new diagram.)
The question comes from understanding a Seiberg-Witten theory related to Khovanov homology.
You could also ask similar questions for null-homologous knots in an arbitrary 3-manifold.  (You need null-homologous, since otherwise there's clearly never a pseudo-diagram.)  If we could answer this positively, I think it would allow defining a Khovanov homology theory for null-homologous knots in general.
Best,
        Dylan