SSL Notes for meeting #9 (10/7)

Today's note-taker: Paul (on deck: Sam) 

Today's snack-provider: Carl (on deck: Hal)

Jim - Change in plans: (1) Instead of telling what we have done, we will allot
5 minutes per person to teach or learn items that they are interested in
(2) Tuesday's meetings will take place from 2:30-4:30 for the rest of the 
semester.

Go around with 5 minutes

Sam - Worked with hexagonal snakes and tried to create a bijection between 
the square and hexagonal snakes taking weights into account.  Worked on 
webpage and Maple from Wilf's book.  Asks about matrices that Jim wrote 
on the board last time.

Emily - a=(1 1)  b=(0 1)  ab=(1 2)
          (1 0)    (1 1)     (0 1)
If we start with a box (call it e) and go to the right, choose either a or b 
to represent going to the right, and the nonchosen matrix represents taking a 
turn upward.  For example, if we have a snake going right, right, up, up, 
right, right, right, that corresponds to the matrix eaabbaaa.  If we multiply 
by matrix multiplication, the sum of the entries of eaabbaaa will correspond 
to the number of perfect matchings in that snake.

Paul - Read that every markoff number could be found by multiplying two 
matrices, namely 
A=(1 1) B=(2 1)
  (1 0)   (1 0)
in certain ways to obtain a markoff number in the upper left entry of that 
matrix.  Call C=A^2 and D=B^2.  Then the markoff numbers correspond to the 
upper left entry of matrices that start and end with B and have a string of 
palindromes (eg DDCDCDD) between the B's.  Used that and the fact that every 
other fibonacci number as well as every other pellian is a markoff number to 
start looking for combinatorial representations of markoff numbers. the 
fibonacci snake is known. the pell snake is a snake with three boxes 
horizontal, followed by three vertical, followed by three horizontal, etc.  
The pell numbers are the corner boxes' perfect matchings. There has to be 
some way to combine the pell and fibonacci graph to obtain more markoff 
numbers.

Jim - Markoff numbers can be found by beginning with (1,1,1) and changing 
only one number in it to the sum of the squares of the unchanged numbers 
divided by the (about to be) changed 
number.

Carl - Thought he found a non integer in the sequence and recurrence that Jim 
gave us via email.  Learned later that he used 3766 instead of 37666 to 
compute the incorrect number. Elaborated on Emily's ideas of A and B 
representing the snake.  Showed how we can use matrix representation to 
splice and glue snakes together in a nonoverlapping fasion.

Hal - Thinking of answers to Jim's question that he had sent over email.  
How is this not a cluster algebra?

Jim - This becomes a cluster algebra with the right selection of variables.

Hal - It's concievable that it would not be just a tree.

Stephen - In a general cluster algebra, it is not just a tree.

Jim - I can see five ways to put a snake together.  Let G and H be snakes.
     (1) Right edge of G's last box with Left edge of H's first box.
     (2) Top edge of G's last box with Bottom edge of H's first box.
     (3) Superimposing G's last box on top of H's first box.
     (4&5) Drawing two new edges between the two in two ways.

Sam - We could superimpose two boxes.

Carl - Superimposition of 2-squares may lose some information.

Jim - It wouldn't actually lose information.

Emily - noticed in putting two snakes together, even though the sums of the 
entries of bbbaa and aaabb are the same, when connected with another graph, 
they do not, in general, produce the same sums.

Jim - We believe adding entries of matrices is counting the perfect matchings.
See if we can see what they actually count.

Abby - Worked on slanted boxes and expanded a and b to higher dimensions.  
(eg a(3)=(1 1 1)
         (1 0 0)
         (1 0 0)
b(3)=(0 0 1)
     (0 0 1)
     (1 1 1)
and found that abaab in three dimensions had some interesting properties.  
The matrix abaab had the property that the (1,1) entry times the (3,3) entry 
plus the (3,3) entry minus 1 gave the (1,3) entry.  Similar properties were 
found in 2 and 4 dimensional matrices.

Paul - Could extending these to three dimensions tell us anything about 
perfect matchings in three dimensions (i.e. cube perfect matchings)?

Jim - The pellian numbers gives a good approximation to square root of 2, 
similar to how the fibonacci sequence gives a good approximation to the 
golden ratio.

(1/1), (3/2), (7/5), (17/12), (41/29), (99/70)... gets close to root two.

What is the benefit and drawback to setting matrix a to mean right or up as 
opposed to keeping generality between a and b?

Martin - I like canonical definitions so its ok to set a and b.

Jim -  The issue of whether we want to use a, b fixed or a, b free is like the
square root problem in algebra.  In complex values, there is an ambiguity of 
square roots, in the real numbers, there is not.  So we would want that 
ambiguity if we were working with complex numbers.

Time for a break, then I will show you some research problems.

Relocate to B107.

Problems:
(1) Why Laurent polynomials???
(2) What do the coefficients in these Laurent polynomials mean?
(3) What is the relationship between successive operons in the numerators of 
these polynomials?

See the link to after.mws on http://jamespropp.org/SSL/ for more 
information.