Meeting Thursday Oct. 30, 2003
Note-taker:  Sam
Snacks: Paul

Hal will take notes next time.
Emilie will bring snacks next time.

Hal: Stephen, could you explain your suggestion about simplifying the calculations I 
was doing in Maple regarding the Soddy equation and the Apollonian gasket?

Stephen: Sure. What you want to do is to use four variables instead of three. [Stephen 
draws the tangent circles and writes Soddy's equation 2(a^2+b^2+c^2+d^2) = (a+b+c+d)^2]

[Hal will work on this during the meeting and report back]

Stephen: One we thing we still need to resolve is the assertion in the Itsara article 
which Sam brought up on Tuesday that an arbitrary snake can be transformed into a 
2-by-k straight snake for some k, with only single and double edges. 

Martin: Do we believe this to be true?
Paul: We do not currently have a way of doing it.

[Hal goes up to the board. He has an idea of extending snakes instead of collapsing 
them. It amounts to having straight snakes with double, single, or *zero* edges.]
Hal: Clearly doesn't match up with claim, but might be useful somehow.

Martin: I'd also like to work on the alternating pattern in the matchings of the 
2-by-2-by-n snakes.
Emilie: I have been working on Jim's idea, where we split a triangle into six and have 
some way of ascribing rational values to the sides. I have a system of nine (non-linear)
equations that I am going to try to solve in Maple.

Stephen: We'll work in groups or individually and then reconvene at ten till to 
summarize for the minutes.

[At 5:20 . . .]
Stephen: I'll start. I mostly talked with Sam about the Itsara assertion, and I read 
some of Cluster Algebras II. 
Abby: Talked with Carl about what he has been working on - matchings of faces of cubes. 
Also talked with Paul about some possible bijections (or even well-defined maps) of 
2-by-2-by-n snake graph matchings and pairs of tilings of the 3-by-2n grid.
Downloaded graph.tcl and got it to work sufficiently well.

Martin: Worked with Hal and his Apollonian gasket cluster algebra equations in Maple. 
Worked with Paul a little also on the above. Perhaps my previous bijection between 
partial sums of the one sequence (1,3,11,41,153,...) and the other sequence 
(1,4,15,56,209,...) could be of some help.

Hal: Worked on the Apollonian gasket and programming in Maple, I still ended up with 
lots of nested radicals. 
[Stephen interjects and draws Soddy's equation on the board]
Stephen: It's like the Markoff recurrence where we have been substituting. 
Hal: Oh, I see it. I'll have something for next time for sure.

Sam: Worked with Stephen. Realized a few things. First, the assertion is not true for 
arbitrary snakes, in fact, it is not true for "staircase snakes" at all (that is, 
snakes with n boxes and n+1 matchings). Martin was able to show exhaustively that 
there is no such configuartion of a straight rectangle with double and single edges 
having six perfect matchings.
Second, Stephen and I sketched a proof of why this identifying vertices around a 
corner maneuver gives the same number of matchings as the original graph. I can 
write this up. What's left is to understand why the graphs in question in the Itsara 
article are never staircases (or never even have staircase pieces).

Emilie: Worked on my system of equations.
Carl: Conversed with everyone and played around a bit with matchings of faces of cubes.