Sum of Dynnikov coords

[jeanluct]

In issue #17 we observed the following: consider the three loops

#!matlab
   (( 0 -1  1 -1  0 -1 ))
   (( 1 -1  2 -1  0 -3 ))
   (( 1  0  1  0  0 -2 ))

Claim: the middle one is the sum of the other two, but it is also their union! When does this happen? Clearly they need to be disjoint.

So is this a theorem? Given Dynnikov coordinates for two disjoint loops, the sum of the coordinates is the coordinates of their union.

Marko and I in a discussion thought we could generalize this to loops that intersect by resolving crossings using some orientation on the loops. It seemed to work, but I think it is easy to come up with a counterexample:

#!matlab

l1 = loop([0 0 -1 1]);
l2 = loop([1 -2 -1 0]);

subplot(2,1,1)
plot(l1,'PunctureSize',.02)
hold on
plot(l2,'LineColor','r','PunctureSize',0)
hold off
subplot(2,1,2)
plot(loop(l1.coords+l2.coords))

[jeanluct]

From Jean-Luc Thiffeault on 2014-10-11 16:31:36+00:00

Putting on hold. Maybe we'll get back to it sometime.

[jeanluct]

BitBucket's "On hold" is not a status recognized by GitHub. Closing instead. Use filtering to find later.