MEANDERS

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MEANDERS
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Who Cares??? (besides combinatorists?)

• Meanders show up in several places including
• Study of Polymers (unusually long molecules) and
  their compact foldings
• Planar Algebras
• Statistical Mechanics
• Sorting Jordan Sequences
• Matrix Models
• Defects in Liquid Crystals
• Dimensional Gravity

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WHAT IS A MEANDER???

• Actually there are several types
• Open Meanders
• Closed Meanders
• Semi Meanders

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WHAT IS A MEANDER???

• Actually there are several types
• Open Meanders
• Closed Meanders
• Semi Meanders

Definition An open meander of order n is a self
avoiding curve that travels from left to
right crossing an infinite horizontal
line n times. The number of open
meanders of order n, unique up to
homeomorphism, is denoted mn
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WHAT IS A MEANDER???

• Actually there are several types
• Open Meanders
• Closed Meanders
• Semi Meanders

Definition A closed meander of order n is a
closed self avoiding curve that
crosses an infinite horizontal line 2n times.
The number of closed meanders of order n,
unique up to homeomorphism, is denoted
Mn
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WHAT IS A MEANDER???

• Actually there are several types
• Open Meanders
• Closed Meanders
• Semi Meanders

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• Important Background
• Meanders are both interesting (and difficult) to
  count because they combine ideas of topology and
  combinatorics
• Topology is a mathematical branch that studies
  spatial properties and how they behave under
  continuous deformations
• Homeomorphism ? Homomorphism
• Homeomorphism roughly means topological
  isomorphism
• i.e. Two shapes are Homeomorphic if they
  can be continuously (no cutting) deformed into
  one another

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Making Life Easier
Clearly trying to count meanders, up to
equivalence, using solely the definition of
homeomorphism would be painful
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Making Life Easier
Clearly trying to count meanders, up to
equivalence, using solely the definition of
homeomorphism would be painful
What we want is an easier way to categorize the
meanders!!!
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Canonical Form for Closed Meanders

• For closed meanders we can define a canonical
  form which is an ordered pair of arch formations
• An arch formation of order n is just some group
  of n semi-circles as pictured below

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Canonical Form for Closed Meanders

• For an ordered pair (A1,A2) of arch formations
  A1 may denote the arch formation of the upper
  half of the meander, and A2 the arch formation
  for the lower half... this is the canonical form!

• The point Every homeomorphic (closed) meander
  will have the same canonical form... (need to
  prove)

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Canonical Form for Closed Meanders

• Somewhat obvious fact every closed meander, of
  order n, corresponds to some ordered pair of arch
  formations with n semi-circles in both the upper
  and lower halves.
• Very obvious fact Not every pair of arch
  formations leads to a closed meander, for
  instance if you pair an arch formation with
  itself in most cases it wont lead to a meander

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First attempts at Counting
Fact Mn m2n-1 Proof (by picture) clearly an
open meander that crosses the line an odd number
of times will have its loose ends on opposite
sides of the line, connecting these loose ends
adds an extra crossing (2n-112n) and it also
closes the curve creating a unique closed meander
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First attempts at Counting
in Fact Mn m2n-1 Since breaking the loop of
a closed meander at the right hand side leads
uniquely to an open meander
Also for every open meander counted by m2n we can
close it uniquely to get a closed meander... but
the relationship isnt bijective, we can take a
close meander and open it into the form m2n in
many ways...
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First attempts at Counting
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First attempts at Counting
Note Since closed meanders are easier to count
due to their canonical forms, and since they
provide bounds on the number of open meanders...
for convenience most combinatorists concentrate
on counting closed meanders....
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First attempts at Counting
Lets see if we can use the cannonical forms to
get a bound on Mn

• First we can encode the arch form as a string of
  parenthesis
• Every time we encounter the start of a
  semi-circle write down an (
• When we end a semi-circle put down a )

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First attempts at Counting
Lets see if we can use the cannonical forms to
get a bound on Mn

• Clearly if the semi-circles dont cross each
  other then the string of parenthesis will be well
  formed
• Each 2n-string of well-formed parenthesis
  corresponds uniquely to some arch formation of
  order n
• The number of 2n-strings of well formed
  parenthesis is known to be counted by Cn the nth
  catalan number
• Therefore Since each arch formation, when paired
  with some other arch formation, leads to a closed
  meander we get
• Mn Cn

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First attempts at Counting
Lets see if we can use the cannonical forms to
get a bound on Mn

• Also since every closed meander is some orderded
  pair of arch formations we see that
• Cn2 Mn

This gives the bound on the number of closed
meanders of order n as
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First attempts at Counting
How do these numbers compare?
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Another Attempt

• If we consider the encoding of meanders as
  parenthesis we could create a language of
  meanders consisting of ordered pairs of
  parenthesis
• We could now try to construct a context-free
  grammar for the language of meanders
• It only has a 4 character alphabet

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Another Attempt

• We can develope several production rules but we
  quickly run into trouble as the grammar doesnt
  appear to be free from context, that is the
  preceding choice of rules limits the rules we can
  use in the future.

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Another Attempt
The previous idea does lead into a better idea
though, a Transfer Matrix approach by I. Jensen.
The idea uses the previous alphabet but accepts
there are no simple production rules.

• Idea is to have a boundary line that sweeps
  across the vertical infinite line, and considers
  all possible extentions of the meander
• Each move of the boundary line could add a
  crossing of the infinite line in two ways
• By putting in a new new loop the pairs (,(
  or ),)
• or by dragging a current loop end above the line
  the pairs (,) or ),(

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Another Attempt
The previous idea does lead into a better idea
though, a Transfer Matrix approach by I. Jensen
An example of how this approach would work
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Another Attempt
The previous idea does lead into a better idea
though, a Transfer Matrix approach by I. Jensen
But we must be careful not all moves are valid
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Transfer Matrix Method
Here is the essence of the algorithm

• First for convenience of the computer, we encode
  the current state of the boundary line by a set
  of 0s and 1s, where the number of 1s never
  exceeds the number of 0s.
• This is equivalent to well-formed parenthesis,
  where ( 0 and ) 1
• The difference from before is that we are looking
  at arch ends vertically instead of horizontally,
  to differentiate the two we will call them loop
  ends, when viewed vertically.

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Transfer Matrix Method
Here is the essence of the algorithm
Here is an example of the vertical encoding of
loop ends
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Transfer Matrix Method
Here is the essence of the algorithm

• At every step of the algorithm we maintain a pair
  of integers (h,S) --where h is the number of loop
  ends below the infinite line and S is the binary
  encoding of the loop ends (Dyck Word) more
  pricecly S b0b1b2...bn
• Start by setting n max Mn ( N) we wish to
  calculate
• Initialize a set Sig (h,S) (1, 01)
• Set for each element of Sig count 1
• Set Mn 1
• Set num_crossings 1

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Transfer Matrix Method
Here is the essence of the algorithm

• While num_crossings lt 2N -1 repeat the following
  two steps
• Move the boundary line ahead one step and add a
  crossing to all elements of the Set Sig i.e.
  set num_crossings num_crossings 1
• if num_crossings is odd set j
  (num_crossings1)/2 remove from set Sig the
  element (1,01) and return its count as Mj

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Transfer Matrix Method
Here is the essence of the algorithm

• But how do we do step 1 (adding a crossing)?
• Again there are two ways to add a crossing
• 1-1 By adding a new loop
• 1-2 By draging a loop end to the other side of
  the line

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Transfer Matrix Method
Here is the essence of the algorithm

• 1-1 Adding a new loop

• To do this operations...
• For each element (h,S1S2) of the set Sig
  where
• S1 is the part of the encoding for the loop
  ends below the line
• We add a different element to Sig equal to
  (h1,S101S2)
• And the count of this element is equal to the
  count of the old element

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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2 Dragging

• This operation is even more complex as there are
  two more sub cases, since when we drag the
  loop-end across the infinite line we additionally
  have the option of connecting it to another loop
  end or not. The two cases are then
• 1-2-No Connection
• 1-2-Connection

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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-No Connection

• For each element of Sig of the form (h, S)
  including the element we tossed out in step 1-1
  we remove it from Sig and add two new elements
• (h-1,S)
• (h1,S)
• These operations are allowed provided that we
  dont get that h-1 lt 0 or that h1 isnt
  greater then twice the number of 1s in S

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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

For each element (h,S) of Sig which we removed in
the previous step we additionally consider 4
different casses of adding a connection We
consider these four additional cases in turn.
(Note only one of them will apply)
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case I -00 We Connect a 0-loop end below the
line to a 0-loop end above the line this is
reflected by adding for an element of the form
(h, S100S2) of Sig a new element (h-1,S1S2)
where S2 S2 but since now the of 1s
exceeds of 0s, we change the first 1, in the
sequence that causes this to a 0. E.g. 000111
-gt 0 111 -gt 0 011 -gt 0011
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case I -00
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case II -10 We Connect a 1-loop end from below
to a 0-loop end above. This is easily acheived by
replacing an element of the form (h,S110S2) by
(h-1,S1S2)
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case II -10
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case III -01 This is the case where we connect a
0-loop end from below to a 1-loop end from
above... but this clearly closses of a component
so we dont do this set, unless we have only two
loop ends left and we are actually finishing the
meander... but this is handeled in step 2 of the
algorithm
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case IV -11 This case is connecting a 1-loop end
from below to a 1-loop end above. For this we
replace (h,S111S2) by (h-1, S1S2) Where S1
is computed the same way as S2 from Case I
except in reverse ie (h,001011)-gt(h,0011)
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Transfer Matrix Method
Here is the essence of the algorithm

• 1-2-Connection

Case IV -11
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Transfer Matrix Method
Here is the essence of the algorithm

• Notes on the Algorithm
• For each element we remove from Sig we could
  potentially replace it by 4 new ones
• Adding a new loop (step 1-1)
• Dragging the loop end across from below
• Dragging the loop end across from above
• Making one of 4 types of Connections
• The counts for the new elements we generate are
  calculated by summing over the counts of all
  elements that could potentially have generated
  it.

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THE END