|
|
Line 24: |
Line 24: |
|
x}{4725}-\frac{t_2^7 x}{4725}+\frac{1}{675} t_1 t_2^6 x-\frac{1}{225} t_1^2 t_2^5 x+\frac{1}{135} t_1^3 t_2^4 x-\frac{1}{135} |
|
x}{4725}-\frac{t_2^7 x}{4725}+\frac{1}{675} t_1 t_2^6 x-\frac{1}{225} t_1^2 t_2^5 x+\frac{1}{135} t_1^3 t_2^4 x-\frac{1}{135} |
|
t_1^4 t_2^3 x+\frac{1}{225} t_1^5 t_2^2 x-\frac{1}{675} t_1^6 t_2 x</math> |
|
t_1^4 t_2^3 x+\frac{1}{225} t_1^5 t_2^2 x-\frac{1}{675} t_1^6 t_2 x</math> |
|
|
:<math>-\frac{1}{11} t_1 x^9+\frac{t_2 |
|
|
x^9}{11}+\frac{2414 t_1^3 x^7}{51975}-\frac{2414 t_2^3 x^7}{51975}+\frac{53243 t_1 t_2^2 |
|
|
x^7}{467775}-\frac{53243 t_1^2 t_2 x^7}{467775}</math> |
|
|
::<math>-\frac{4058 t_1^5 x^5}{467775}+\frac{4058 t_2^5 |
|
|
x^5}{467775}-\frac{3904 t_1 t_2^4 x^5}{93555}+\frac{782 t_1^2 t_2^3 x^5}{10395}-\frac{782 t_1^3 t_2^2 |
|
|
x^5}{10395}+\frac{3904 t_1^4 t_2 x^5}{93555}</math> |
|
|
::<math>+\frac{331 t_1^7 x^3}{467775}-\frac{331 t_2^7 |
|
|
x^3}{467775}+\frac{331 t_1 t_2^6 x^3}{66825}-\frac{331 t_1^2 t_2^5 x^3}{22275}+\frac{331 t_1^3 t_2^4 |
|
|
x^3}{13365}-\frac{331 t_1^4 t_2^3 x^3}{13365}+\frac{331 t_1^5 t_2^2 x^3}{22275}-\frac{331 t_1^6 t_2 |
|
|
x^3}{66825}</math> |
|
|
::<math>-\frac{2 t_1^9 x}{93555}+\frac{2 t_2^9 x}{93555}-\frac{2 t_1 t_2^8 x}{10395}+\frac{8 t_1^2 |
|
|
t_2^7 x}{10395}-\frac{8 t_1^3 t_2^6 x}{4455}+\frac{4 t_1^4 t_2^5 x}{1485}-\frac{4 t_1^5 t_2^4 |
|
|
x}{1485}+\frac{8 t_1^6 t_2^3 x}{4455}-\frac{8 t_1^7 t_2^2 x}{10395}+\frac{2 t_1^8 t_2 |
|
|
x}{10395}</math> |
|
|
|
|
|
and |
|
and |
Line 31: |
Line 45: |
|
:<math>-\frac{1147 t_1^2 x^6}{14175}+\frac{1147 t_1 t_2 |
|
:<math>-\frac{1147 t_1^2 x^6}{14175}+\frac{1147 t_1 t_2 |
|
x^6}{14175}+\frac{13}{525} t_1^4 x^4+\frac{878 t_1^2 t_2^2 x^4}{14175}-\frac{1229 t_1^3 t_2 x^4}{14175}-\frac{1}{525} t_1^6 |
|
x^6}{14175}+\frac{13}{525} t_1^4 x^4+\frac{878 t_1^2 t_2^2 x^4}{14175}-\frac{1229 t_1^3 t_2 x^4}{14175}-\frac{1}{525} t_1^6 |
|
x^2+\frac{2}{105} t_1^3 t_2^3 x^2-\frac{1}{35} t_1^4 t_2^2 x^2+\frac{2}{175} t_1^5 t_2 x^2</math>? |
|
x^2+\frac{2}{105} t_1^3 t_2^3 x^2-\frac{1}{35} t_1^4 t_2^2 x^2+\frac{2}{175} t_1^5 t_2 x^2</math> |
|
|
:<math>-\frac{2939 t_1^2 |
|
|
x^8}{42525}+\frac{2939 t_1 t_2 x^8}{42525}+\frac{1327 t_1^4 x^6}{42525}+\frac{2896 t_1^2 t_2^2 |
|
|
x^6}{42525}-\frac{4223 t_1^3 t_2 x^6}{42525}</math> |
|
|
::<math>-\frac{199 t_1^6 x^4}{42525}+\frac{20}{567} t_1^3 t_2^3 |
|
|
x^4-\frac{97 t_1^4 t_2^2 x^4}{1701}+\frac{1124 t_1^5 t_2 x^4}{42525}+\frac{2 t_1^8 |
|
|
x^2}{8505}+\frac{2}{243} t_1^4 t_2^4 x^2-\frac{16 t_1^5 t_2^3 x^2}{1215}+\frac{8 t_1^6 t_2^2 |
|
|
x^2}{1215}-\frac{16 t_1^7 t_2 x^2}{8505}</math>? |
|
|
|
|
|
==A HOMFLY Braidor== |
|
==A HOMFLY Braidor== |
Revision as of 22:02, 23 January 2007
A Second Look
We'll take a second look at 06-1350/Class Notes for Tuesday October 24.
A Numerology Problem
Question. Can you find nice formulas for the functions
and
of the variables
,
and
, whose Taylor expansions are
![{\displaystyle -{\frac {1}{5}}t_{1}x^{3}+{\frac {t_{2}x^{3}}{5}}+{\frac {t_{1}^{3}x}{45}}-{\frac {t_{2}^{3}x}{45}}+{\frac {1}{15}}t_{1}t_{2}^{2}x-{\frac {1}{15}}t_{1}^{2}t_{2}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5704ac5ae947757b489e7e7867fa0122dbc4f806)
![{\displaystyle -{\frac {2t_{1}^{5}x}{945}}+{\frac {2t_{2}^{5}x}{945}}-{\frac {2}{189}}t_{1}t_{2}^{4}x+{\frac {4}{189}}t_{1}^{2}t_{2}^{3}x-{\frac {4}{189}}t_{1}^{3}t_{2}^{2}x+{\frac {2}{189}}t_{1}^{4}t_{2}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36fe6b5b92274ae6f45381dba98c4f6aad91556d)
![{\displaystyle -{\frac {74t_{1}^{5}x^{3}}{14175}}+{\frac {74t_{2}^{5}x^{3}}{14175}}-{\frac {74t_{1}t_{2}^{4}x^{3}}{2835}}+{\frac {148t_{1}^{2}t_{2}^{3}x^{3}}{2835}}-{\frac {148t_{1}^{3}t_{2}^{2}x^{3}}{2835}}+{\frac {74t_{1}^{4}t_{2}x^{3}}{2835}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc3c59a4764d53af06eb1773796b82a2932699f)
![{\displaystyle +{\frac {t_{1}^{7}x}{4725}}-{\frac {t_{2}^{7}x}{4725}}+{\frac {1}{675}}t_{1}t_{2}^{6}x-{\frac {1}{225}}t_{1}^{2}t_{2}^{5}x+{\frac {1}{135}}t_{1}^{3}t_{2}^{4}x-{\frac {1}{135}}t_{1}^{4}t_{2}^{3}x+{\frac {1}{225}}t_{1}^{5}t_{2}^{2}x-{\frac {1}{675}}t_{1}^{6}t_{2}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52841a01cd7b418eae83d1c2dca1011b56654992)
![{\displaystyle -{\frac {4058t_{1}^{5}x^{5}}{467775}}+{\frac {4058t_{2}^{5}x^{5}}{467775}}-{\frac {3904t_{1}t_{2}^{4}x^{5}}{93555}}+{\frac {782t_{1}^{2}t_{2}^{3}x^{5}}{10395}}-{\frac {782t_{1}^{3}t_{2}^{2}x^{5}}{10395}}+{\frac {3904t_{1}^{4}t_{2}x^{5}}{93555}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d4acad7a54db94fd98ef021f01addde6f13f47)
![{\displaystyle +{\frac {331t_{1}^{7}x^{3}}{467775}}-{\frac {331t_{2}^{7}x^{3}}{467775}}+{\frac {331t_{1}t_{2}^{6}x^{3}}{66825}}-{\frac {331t_{1}^{2}t_{2}^{5}x^{3}}{22275}}+{\frac {331t_{1}^{3}t_{2}^{4}x^{3}}{13365}}-{\frac {331t_{1}^{4}t_{2}^{3}x^{3}}{13365}}+{\frac {331t_{1}^{5}t_{2}^{2}x^{3}}{22275}}-{\frac {331t_{1}^{6}t_{2}x^{3}}{66825}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1acdbf35fd57c9037942ae933c8d9e0d070bb307)
![{\displaystyle -{\frac {2t_{1}^{9}x}{93555}}+{\frac {2t_{2}^{9}x}{93555}}-{\frac {2t_{1}t_{2}^{8}x}{10395}}+{\frac {8t_{1}^{2}t_{2}^{7}x}{10395}}-{\frac {8t_{1}^{3}t_{2}^{6}x}{4455}}+{\frac {4t_{1}^{4}t_{2}^{5}x}{1485}}-{\frac {4t_{1}^{5}t_{2}^{4}x}{1485}}+{\frac {8t_{1}^{6}t_{2}^{3}x}{4455}}-{\frac {8t_{1}^{7}t_{2}^{2}x}{10395}}+{\frac {2t_{1}^{8}t_{2}x}{10395}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe0f4657619a0eb655ab9e7140f707bf7eeb37b)
and
![{\displaystyle -{\frac {1147t_{1}^{2}x^{6}}{14175}}+{\frac {1147t_{1}t_{2}x^{6}}{14175}}+{\frac {13}{525}}t_{1}^{4}x^{4}+{\frac {878t_{1}^{2}t_{2}^{2}x^{4}}{14175}}-{\frac {1229t_{1}^{3}t_{2}x^{4}}{14175}}-{\frac {1}{525}}t_{1}^{6}x^{2}+{\frac {2}{105}}t_{1}^{3}t_{2}^{3}x^{2}-{\frac {1}{35}}t_{1}^{4}t_{2}^{2}x^{2}+{\frac {2}{175}}t_{1}^{5}t_{2}x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90bf21c1cb7fa3e09e2f9b1463124c8b8b47e446)
?
A HOMFLY Braidor
The Algebra
Let
be the free associative (but non-commutative) algebra generated by the elements of the symmetric group
on
and by formal variables
and
, and let
be the quotient of
by the following "HOMFLY" relations:
commutes with everything else.
- The product of permutations is as in the symmetric group
.
- If
is a permutation then
.
, where
is the transposition of
and
.
Finally, declare that
while
for every
and every
, and let
be the graded completion of
.
We say that an element of
is "sorted" if it is written in the form
where
is a permutation and
and the
's are all non-negative integer. The HOMFLY relations imply that every element of
is a linear combinations of sorted elements. Thus as a vector space,
can be identified with the ring
of power series in the variables
tensored with the group ring of
. The product of
is of course very different than that of
.
Examples.
- The general element of
is
where
denotes the identity permutation and
is a power series in two variables
and
.
is commutative.
- The general element of
is
where
and
are power series in three variables and
and
are the two elements of
.
is not commutative and its product is non-trivial to describe.
- The general element of
is described using
power series in 4 variables. The general element of
is described using n! power series in
variables.
The algebra
embeds in
in a trivial way by regarding
as a subset of
in the obvious manner; thus when given an element of
we are free to think of it also as an element of
. There is also a non-trivial map
defined as follows:
.
.
acts on permutations by "shifting them one unit to the right", i.e., by identifying
with
.
The Equations
We seek to find a "braidor"; an element
of
satisfying:
(higher order terms).
in
.
With the vector space identification of
with
in mind, we are seeking two power series of three variables each, whose low order behaviour is specified and which are required to satisfy 6 functional equations written in terms of 4 variables.
The Equations in Functional Form
A Solution
The first few terms of a solution can be computed using a computer, as shown above and below. But a true solution, written in a functional form, is still missing.
Computer Games
A primitive mathematica program to play with these objects is here.