# The HOMFLY Braidor Algebra

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In Preparation

The information below is preliminary and cannot be trusted! (v)

This paperlet is about yet another construction of the HOMFLY polynomial, this time using "braidor equations". Though at the moment the term "braidor equations", the relationship with HOMFLY and the rationale for the whole plan is not yet described here. If you know what this is about, good. If not, bummer.

## The Algebra

Let ${\displaystyle A_{n}^{0}=\langle S_{n},x,t_{1},\ldots t_{n}\rangle }$ be the free associative (but non-commutative) algebra generated by the elements of the symmetric group ${\displaystyle S_{n}}$ on ${\displaystyle \{1,\ldots ,n\}}$ and by formal variables ${\displaystyle x}$ and ${\displaystyle t_{1}\ldots t_{n}}$, and let ${\displaystyle A_{n}^{1}}$ be the quotient of ${\displaystyle A_{n}^{0}}$ by the following "HOMFLY" relations:

1. ${\displaystyle x}$ commutes with everything else.
2. The product of permutations is as in the symmetric group ${\displaystyle S_{n}}$.
3. If ${\displaystyle \sigma }$ is a permutation then ${\displaystyle t_{i}\sigma =\sigma t_{\sigma i}}$.
4. ${\displaystyle [t_{i},t_{j}]=x\sigma _{ij}(t_{i}-t_{j})}$, where ${\displaystyle \sigma _{ij}}$ is the transposition of ${\displaystyle i}$ and ${\displaystyle j}$.

Finally, declare that ${\displaystyle \deg x=\deg t_{i}=1}$ while ${\displaystyle \deg \sigma =0}$ for every ${\displaystyle 1\leq i\leq n}$ and every ${\displaystyle \sigma \in S_{n}}$, and let ${\displaystyle A_{n}}$ be the graded completion of ${\displaystyle A_{n}^{1}}$.

We say that an element of ${\displaystyle A_{n}}$ is "sorted" if it is written in the form ${\displaystyle x^{k}\cdot \sigma t_{1}^{k_{1}}t_{2}^{k_{2}}\cdots t_{n}^{k_{n}}}$ where ${\displaystyle \sigma }$ is a permutation and ${\displaystyle k}$ and the ${\displaystyle k_{i}}$'s are all non-negative integer. The HOMFLY relations imply that every element of ${\displaystyle A_{n}}$ is a linear combinations of sorted elements. Thus as a vector space, ${\displaystyle A_{n}}$ can be identified with the ring ${\displaystyle B_{n}}$ of power series in the variables ${\displaystyle x,t_{1},\ldots ,t_{n}}$ tensored with the group ring of ${\displaystyle S_{n}}$. The product of ${\displaystyle A_{n}}$ is of course very different than that of ${\displaystyle B_{n}}$.

Examples.

1. The general element of ${\displaystyle A_{1}}$ is ${\displaystyle (1)f(x,t_{1})}$ where ${\displaystyle (1)}$ denotes the identity permutation and ${\displaystyle f(x,t_{1})}$ is a power series in two variables ${\displaystyle x}$ and ${\displaystyle t_{1}}$. ${\displaystyle A_{1}}$ is commutative.
2. The general element of ${\displaystyle A_{2}}$ is ${\displaystyle (12)f(x,t_{1},t_{2})+(21)g(x,t_{1},t_{2})}$ where ${\displaystyle f}$ and ${\displaystyle g}$ are power series in three variables and ${\displaystyle (12)}$ and ${\displaystyle (21)}$ are the two elements of ${\displaystyle S_{2}}$. ${\displaystyle A_{2}}$ is not commutative and its product is non-trivial to describe.
3. The general element of ${\displaystyle A_{3}}$ is described using ${\displaystyle 3!=6}$ power series in 4 variables. The general element of ${\displaystyle A_{n}}$ is described using n! power series in ${\displaystyle n+1}$ variables.

The algebra ${\displaystyle A_{n}}$ embeds in ${\displaystyle A_{n+1}}$ in a trivial way by regarding ${\displaystyle \{1,\ldots ,n\}}$ as a subset of ${\displaystyle \{1,\ldots ,n+1\}}$ in the obvious manner; thus when given an element of ${\displaystyle A_{n}}$ we are free to think of it also as an element of ${\displaystyle A_{n+1}}$. There is also a non-trivial map ${\displaystyle \Delta :A_{n}\to A_{n+1}}$ defined as follows:

1. ${\displaystyle \Delta (x)=x}$.
2. ${\displaystyle \Delta (t_{i})=t_{i+1}+x\sigma _{1,i+1}}$.
3. ${\displaystyle \Delta }$ acts on permutations by "shifting them one unit to the right", i.e., by identifying ${\displaystyle \{1,\ldots ,n\}}$ with ${\displaystyle \{2,\ldots ,n+1\}\subset \{1,\ldots ,n+1\}}$.

## The Equations

We seek to find a "braidor"; an element ${\displaystyle B}$ of ${\displaystyle A_{2}}$ satisfying:

• ${\displaystyle B=(21)+x(12)+}$(higher order terms).
• ${\displaystyle B(\Delta B)B=(\Delta B)B(\Delta B)}$ in ${\displaystyle A_{3}}$.

With the vector space identification of ${\displaystyle A_{n}}$ with ${\displaystyle B_{n}}$ 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

Lemma. The following identities hold in ${\displaystyle A_{n}}$:

1. ${\displaystyle [t_{i}^{k},t_{j}]=x\sigma _{ij}(t_{i}^{k}-t_{j}^{k})}$ and therefore ${\displaystyle [e^{\alpha t_{i}},t_{j}]=x\sigma _{ij}(e^{\alpha t_{i}}-e^{\alpha t_{j}})}$.
2. ${\displaystyle [t_{i}^{k},t_{j}^{l}]=x\sigma _{ij}\left({\frac {t_{i}^{k+l}+t_{j}^{k+l}-t_{i}^{k}t_{j}^{l}-t_{i}^{l}t_{j}^{k}}{t_{i}-t_{j}}}\right)}$ and therefore ${\displaystyle [e^{\alpha t_{i}},e^{\beta t_{j}}]=x\sigma _{ij}\left({\frac {e^{(\alpha +\beta )t_{i}}+e^{(\alpha +\beta )t_{j}}-e^{\alpha t_{i}+\beta t_{j}}-e^{\beta t_{i}+\alpha t_{j}}}{t_{i}-t_{j}}}\right)}$
(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables ${\displaystyle x}$, ${\displaystyle t_{i}}$ and ${\displaystyle t_{j}}$, and then the "true" ${\displaystyle x}$, ${\displaystyle t_{i}}$ and ${\displaystyle t_{j}}$ are to be substituted in, in "normal order" - in every monomial the variables are written so that every ${\displaystyle t_{i}}$ occurs before any ${\displaystyle t_{j}}$).
3. ${\displaystyle \Delta (t_{i}^{k})=t_{i+1}^{k}+{\frac {x}{2}}\left({\frac {(t_{i+1}+x)^{k}-t_{1}^{k}}{t_{i+1}+x-t_{1}}}-{\frac {(t_{i+1}-x)^{k}-t_{1}^{k}}{t_{i+1}-x-t_{1}}}\right)+\sigma _{1,i+1}{\frac {x}{2}}\left({\frac {(t_{i+1}+x)^{k}-t_{1}^{k}}{t_{i+1}+x-t_{1}}}+{\frac {(t_{i+1}-x)^{k}-t_{1}^{k}}{t_{i+1}-x-t_{1}}}\right)}$ and therefore ${\displaystyle \Delta (e^{t_{i}})=e^{t_{i+1}}+{\frac {x}{2}}\left({\frac {e^{t_{i+1}+x}-e^{t_{1}}}{t_{i+1}+x-t_{1}}}-{\frac {e^{t_{i+1}-x}-e^{t_{1}}}{t_{i+1}-x-t_{1}}}\right)+\sigma _{1,i+1}{\frac {x}{2}}\left({\frac {e^{t_{i+1}+x}-e^{t_{1}}}{t_{i+1}+x-t_{1}}}+{\frac {e^{t_{i+1}-x}-e^{t_{1}}}{t_{i+1}-x-t_{1}}}\right)}$. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables ${\displaystyle x}$, ${\displaystyle t_{1}}$ and ${\displaystyle t_{i+1}}$, and then the "true" ${\displaystyle x}$, ${\displaystyle t_{1}}$ and ${\displaystyle t_{i+1}}$ are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

## A Solution

The first few terms of a solution can be computed using a computer, as shown 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.

## Numerology Problems

### Exponential Version

Question. Can you find nice formulas for the functions ${\displaystyle f_{12}}$ and ${\displaystyle f_{21}}$ of the variables ${\displaystyle t_{1}}$, ${\displaystyle t_{2}}$ and ${\displaystyle x}$, whose Taylor expansions begin with

${\displaystyle f_{12}=x+{\frac {xt_{2}}{3}}-{\frac {xt_{1}}{3}}+{\frac {x^{3}}{6}}-{\frac {13}{90}}t_{1}x^{3}+{\frac {13t_{2}x^{3}}{90}}+{\frac {t_{1}^{3}x}{45}}-{\frac {t_{2}^{3}x}{45}}+{\frac {1}{15}}t_{1}t_{2}^{2}x-{\frac {1}{15}}t_{1}^{2}t_{2}x}$

${\displaystyle +{\frac {x^{5}}{120}}-{\frac {37t_{1}x^{5}}{7560}}+{\frac {37t_{2}x^{5}}{7560}}+{\frac {31t_{1}^{3}x^{3}}{1890}}-{\frac {31t_{2}^{3}x^{3}}{1890}}+{\frac {31}{630}}t_{1}t_{2}^{2}x^{3}-{\frac {31}{630}}t_{1}^{2}t_{2}x^{3}}$
${\displaystyle -{\frac {2t_{1}^{5}x}{945}}+{\frac {2t_{2}^{5}x}{945}}-{\frac {2}{189}}t_{1}t_{2}^{4}x+{\frac {4}{189}}t_{1}^{2}t_{2}^{3}x-{\frac {4}{189}}t_{1}^{3}t_{2}^{2}x+{\frac {2}{189}}t_{1}^{4}t_{2}x}$
${\displaystyle +{\frac {x^{7}}{5040}}-{\frac {29t_{1}x^{7}}{75600}}+{\frac {29t_{2}x^{7}}{75600}}+{\frac {293t_{1}^{3}x^{5}}{113400}}-{\frac {293t_{2}^{3}x^{5}}{113400}}-{\frac {521t_{1}t_{2}^{2}x^{5}}{113400}}+{\frac {521t_{1}^{2}t_{2}x^{5}}{113400}}}$
${\displaystyle -{\frac {29t_{1}^{5}x^{3}}{14175}}+{\frac {29t_{2}^{5}x^{3}}{14175}}-{\frac {29t_{1}t_{2}^{4}x^{3}}{2835}}+{\frac {58t_{1}^{2}t_{2}^{3}x^{3}}{2835}}-{\frac {58t_{1}^{3}t_{2}^{2}x^{3}}{2835}}+{\frac {29t_{1}^{4}t_{2}x^{3}}{2835}}}$
${\displaystyle +{\frac {t_{1}^{7}x}{4725}}-{\frac {t_{2}^{7}x}{4725}}+{\frac {1}{675}}t_{1}t_{2}^{6}x-{\frac {1}{225}}t_{1}^{2}t_{2}^{5}x+{\frac {1}{135}}t_{1}^{3}t_{2}^{4}x-{\frac {1}{135}}t_{1}^{4}t_{2}^{3}x+{\frac {1}{225}}t_{1}^{5}t_{2}^{2}x-{\frac {1}{675}}t_{1}^{6}t_{2}x}$
${\displaystyle +{\frac {x^{9}}{362880}}+{\frac {1129t_{1}x^{9}}{59875200}}-{\frac {1129t_{2}x^{9}}{59875200}}-{\frac {743t_{1}^{3}x^{7}}{7484400}}+{\frac {743t_{2}^{3}x^{7}}{7484400}}+{\frac {779t_{1}t_{2}^{2}x^{7}}{831600}}-{\frac {779t_{1}^{2}t_{2}x^{7}}{831600}}}$
${\displaystyle -{\frac {347t_{1}^{5}x^{5}}{623700}}+{\frac {347t_{2}^{5}x^{5}}{623700}}-{\frac {85t_{1}t_{2}^{4}x^{5}}{74844}}-{\frac {223t_{1}^{2}t_{2}^{3}x^{5}}{37422}}+{\frac {223t_{1}^{3}t_{2}^{2}x^{5}}{37422}}+{\frac {85t_{1}^{4}t_{2}x^{5}}{74844}}}$
${\displaystyle +{\frac {233t_{1}^{7}x^{3}}{935550}}-{\frac {233t_{2}^{7}x^{3}}{935550}}+{\frac {233t_{1}t_{2}^{6}x^{3}}{133650}}-{\frac {233t_{1}^{2}t_{2}^{5}x^{3}}{44550}}+{\frac {233t_{1}^{3}t_{2}^{4}x^{3}}{26730}}-{\frac {233t_{1}^{4}t_{2}^{3}x^{3}}{26730}}+{\frac {233t_{1}^{5}t_{2}^{2}x^{3}}{44550}}-{\frac {233t_{1}^{6}t_{2}x^{3}}{133650}}}$
${\displaystyle -{\frac {2t_{1}^{9}x}{93555}}+{\frac {2t_{2}^{9}x}{93555}}-{\frac {2t_{1}t_{2}^{8}x}{10395}}+{\frac {8t_{1}^{2}t_{2}^{7}x}{10395}}-{\frac {8t_{1}^{3}t_{2}^{6}x}{4455}}+{\frac {4t_{1}^{4}t_{2}^{5}x}{1485}}-{\frac {4t_{1}^{5}t_{2}^{4}x}{1485}}+{\frac {8t_{1}^{6}t_{2}^{3}x}{4455}}-{\frac {8t_{1}^{7}t_{2}^{2}x}{10395}}+{\frac {2t_{1}^{8}t_{2}x}{10395}}}$

and

${\displaystyle f_{21}=1+{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {1}{9}}t_{1}^{2}x^{2}+{\frac {1}{9}}t_{1}t_{2}x^{2}}$

${\displaystyle +{\frac {x^{6}}{720}}-{\frac {1}{270}}t_{1}^{2}x^{4}+{\frac {1}{270}}t_{1}t_{2}x^{4}+{\frac {2}{135}}t_{1}^{4}x^{2}+{\frac {2}{45}}t_{1}^{2}t_{2}^{2}x^{2}-{\frac {8}{135}}t_{1}^{3}t_{2}x^{2}}$
${\displaystyle +{\frac {x^{8}}{40320}}-{\frac {41t_{1}^{2}x^{6}}{113400}}+{\frac {41t_{1}t_{2}x^{6}}{113400}}+{\frac {4t_{1}^{4}x^{4}}{1575}}-{\frac {67t_{1}^{2}t_{2}^{2}x^{4}}{14175}}+{\frac {31t_{1}^{3}t_{2}x^{4}}{14175}}-{\frac {1}{525}}t_{1}^{6}x^{2}+{\frac {2}{105}}t_{1}^{3}t_{2}^{3}x^{2}-{\frac {1}{35}}t_{1}^{4}t_{2}^{2}x^{2}+{\frac {2}{175}}t_{1}^{5}t_{2}x^{2}}$
${\displaystyle +{\frac {x^{10}}{3628800}}+{\frac {13t_{1}^{2}x^{8}}{680400}}-{\frac {13t_{1}t_{2}x^{8}}{680400}}-{\frac {17t_{1}^{4}x^{6}}{170100}}+{\frac {53t_{1}^{2}t_{2}^{2}x^{6}}{56700}}-{\frac {71t_{1}^{3}t_{2}x^{6}}{85050}}-{\frac {47t_{1}^{6}x^{4}}{85050}}-{\frac {17t_{1}^{3}t_{2}^{3}x^{4}}{2835}}+{\frac {83t_{1}^{4}t_{2}^{2}x^{4}}{17010}}+{\frac {71t_{1}^{5}t_{2}x^{4}}{42525}}}$
${\displaystyle +{\frac {2t_{1}^{8}x^{2}}{8505}}+{\frac {2}{243}}t_{1}^{4}t_{2}^{4}x^{2}-{\frac {16t_{1}^{5}t_{2}^{3}x^{2}}{1215}}+{\frac {8t_{1}^{6}t_{2}^{2}x^{2}}{1215}}-{\frac {16t_{1}^{7}t_{2}x^{2}}{8505}}}$?

(These Taylor expansions are also available within the mathematica notebook HOMFLY Braidor - Braidor Computations.nb).

### Non-Exponential Version

Question. Can you find nice formulas for the functions ${\displaystyle f'_{12}}$ and ${\displaystyle f'_{21}}$ of the variables ${\displaystyle t_{1}}$, ${\displaystyle t_{2}}$ and ${\displaystyle x}$, whose Taylor expansions begin with

${\displaystyle f'_{12}=x+{\frac {xt_{2}}{3}}-{\frac {xt_{1}}{3}}}$

${\displaystyle -{\frac {1}{5}}t_{1}x^{3}+{\frac {t_{2}x^{3}}{5}}+{\frac {t_{1}^{3}x}{45}}-{\frac {t_{2}^{3}x}{45}}+{\frac {1}{15}}t_{1}t_{2}^{2}x-{\frac {1}{15}}t_{1}^{2}t_{2}x}$
${\displaystyle -{\frac {1}{7}}t_{1}x^{5}+{\frac {t_{2}x^{5}}{7}}+{\frac {11}{315}}t_{1}^{3}x^{3}-{\frac {11}{315}}t_{2}^{3}x^{3}+{\frac {11}{105}}t_{1}t_{2}^{2}x^{3}-{\frac {11}{105}}t_{1}^{2}t_{2}x^{3}}$
${\displaystyle -{\frac {2t_{1}^{5}x}{945}}+{\frac {2t_{2}^{5}x}{945}}-{\frac {2}{189}}t_{1}t_{2}^{4}x+{\frac {4}{189}}t_{1}^{2}t_{2}^{3}x-{\frac {4}{189}}t_{1}^{3}t_{2}^{2}x+{\frac {2}{189}}t_{1}^{4}t_{2}x}$
${\displaystyle -{\frac {1}{9}}t_{1}x^{7}+{\frac {t_{2}x^{7}}{9}}+{\frac {598t_{1}^{3}x^{5}}{14175}}-{\frac {598t_{2}^{3}x^{5}}{14175}}+{\frac {1619t_{1}t_{2}^{2}x^{5}}{14175}}-{\frac {1619t_{1}^{2}t_{2}x^{5}}{14175}}}$
${\displaystyle -{\frac {74t_{1}^{5}x^{3}}{14175}}+{\frac {74t_{2}^{5}x^{3}}{14175}}-{\frac {74t_{1}t_{2}^{4}x^{3}}{2835}}+{\frac {148t_{1}^{2}t_{2}^{3}x^{3}}{2835}}-{\frac {148t_{1}^{3}t_{2}^{2}x^{3}}{2835}}+{\frac {74t_{1}^{4}t_{2}x^{3}}{2835}}}$
${\displaystyle +{\frac {t_{1}^{7}x}{4725}}-{\frac {t_{2}^{7}x}{4725}}+{\frac {1}{675}}t_{1}t_{2}^{6}x-{\frac {1}{225}}t_{1}^{2}t_{2}^{5}x+{\frac {1}{135}}t_{1}^{3}t_{2}^{4}x-{\frac {1}{135}}t_{1}^{4}t_{2}^{3}x+{\frac {1}{225}}t_{1}^{5}t_{2}^{2}x-{\frac {1}{675}}t_{1}^{6}t_{2}x}$
${\displaystyle -{\frac {1}{11}}t_{1}x^{9}+{\frac {t_{2}x^{9}}{11}}+{\frac {2414t_{1}^{3}x^{7}}{51975}}-{\frac {2414t_{2}^{3}x^{7}}{51975}}+{\frac {53243t_{1}t_{2}^{2}x^{7}}{467775}}-{\frac {53243t_{1}^{2}t_{2}x^{7}}{467775}}}$
${\displaystyle -{\frac {4058t_{1}^{5}x^{5}}{467775}}+{\frac {4058t_{2}^{5}x^{5}}{467775}}-{\frac {3904t_{1}t_{2}^{4}x^{5}}{93555}}+{\frac {782t_{1}^{2}t_{2}^{3}x^{5}}{10395}}-{\frac {782t_{1}^{3}t_{2}^{2}x^{5}}{10395}}+{\frac {3904t_{1}^{4}t_{2}x^{5}}{93555}}}$
${\displaystyle +{\frac {331t_{1}^{7}x^{3}}{467775}}-{\frac {331t_{2}^{7}x^{3}}{467775}}+{\frac {331t_{1}t_{2}^{6}x^{3}}{66825}}-{\frac {331t_{1}^{2}t_{2}^{5}x^{3}}{22275}}+{\frac {331t_{1}^{3}t_{2}^{4}x^{3}}{13365}}-{\frac {331t_{1}^{4}t_{2}^{3}x^{3}}{13365}}+{\frac {331t_{1}^{5}t_{2}^{2}x^{3}}{22275}}-{\frac {331t_{1}^{6}t_{2}x^{3}}{66825}}}$
${\displaystyle -{\frac {2t_{1}^{9}x}{93555}}+{\frac {2t_{2}^{9}x}{93555}}-{\frac {2t_{1}t_{2}^{8}x}{10395}}+{\frac {8t_{1}^{2}t_{2}^{7}x}{10395}}-{\frac {8t_{1}^{3}t_{2}^{6}x}{4455}}+{\frac {4t_{1}^{4}t_{2}^{5}x}{1485}}-{\frac {4t_{1}^{5}t_{2}^{4}x}{1485}}+{\frac {8t_{1}^{6}t_{2}^{3}x}{4455}}-{\frac {8t_{1}^{7}t_{2}^{2}x}{10395}}+{\frac {2t_{1}^{8}t_{2}x}{10395}}}$
${\displaystyle -{\frac {1}{13}}t_{1}x^{11}+{\frac {t_{2}x^{11}}{13}}+{\frac {231523t_{1}^{3}x^{9}}{4729725}}-{\frac {231523t_{2}^{3}x^{9}}{4729725}}+{\frac {14046661t_{1}t_{2}^{2}x^{9}}{127702575}}-{\frac {14046661t_{1}^{2}t_{2}x^{9}}{127702575}}}$
${\displaystyle -{\frac {2589746t_{1}^{5}x^{7}}{212837625}}+{\frac {2589746t_{2}^{5}x^{7}}{212837625}}-{\frac {285224t_{1}t_{2}^{4}x^{7}}{5108103}}+{\frac {462340t_{1}^{2}t_{2}^{3}x^{7}}{5108103}}-{\frac {462340t_{1}^{3}t_{2}^{2}x^{7}}{5108103}}+{\frac {285224t_{1}^{4}t_{2}x^{7}}{5108103}}}$
${\displaystyle +{\frac {1304t_{1}^{7}x^{5}}{875875}}-{\frac {1304t_{2}^{7}x^{5}}{875875}}+{\frac {34493t_{1}t_{2}^{6}x^{5}}{3378375}}-{\frac {891986t_{1}^{2}t_{2}^{5}x^{5}}{30405375}}+{\frac {114577t_{1}^{3}t_{2}^{4}x^{5}}{2606175}}-{\frac {114577t_{1}^{4}t_{2}^{3}x^{5}}{2606175}}}$
${\displaystyle +{\frac {891986t_{1}^{5}t_{2}^{2}x^{5}}{30405375}}-{\frac {34493t_{1}^{6}t_{2}x^{5}}{3378375}}}$
${\displaystyle -{\frac {19178t_{1}^{9}x^{3}}{212837625}}+{\frac {19178t_{2}^{9}x^{3}}{212837625}}-{\frac {19178t_{1}t_{2}^{8}x^{3}}{23648625}}+{\frac {76712t_{1}^{2}t_{2}^{7}x^{3}}{23648625}}-{\frac {76712t_{1}^{3}t_{2}^{6}x^{3}}{10135125}}+{\frac {38356t_{1}^{4}t_{2}^{5}x^{3}}{3378375}}-{\frac {38356t_{1}^{5}t_{2}^{4}x^{3}}{3378375}}}$
${\displaystyle +{\frac {76712t_{1}^{6}t_{2}^{3}x^{3}}{10135125}}-{\frac {76712t_{1}^{7}t_{2}^{2}x^{3}}{23648625}}+{\frac {19178t_{1}^{8}t_{2}x^{3}}{23648625}}}$
${\displaystyle +{\frac {1382t_{1}^{11}x}{638512875}}-{\frac {1382t_{2}^{11}x}{638512875}}+{\frac {1382t_{1}t_{2}^{10}x}{58046625}}-{\frac {1382t_{1}^{2}t_{2}^{9}x}{11609325}}+{\frac {1382t_{1}^{3}t_{2}^{8}x}{3869775}}-{\frac {2764t_{1}^{4}t_{2}^{7}x}{3869775}}+{\frac {2764t_{1}^{5}t_{2}^{6}x}{2764125}}-{\frac {2764t_{1}^{6}t_{2}^{5}x}{2764125}}}$
${\displaystyle +{\frac {2764t_{1}^{7}t_{2}^{4}x}{3869775}}-{\frac {1382t_{1}^{8}t_{2}^{3}x}{3869775}}+{\frac {1382t_{1}^{9}t_{2}^{2}x}{11609325}}-{\frac {1382t_{1}^{10}t_{2}x}{58046625}}}$

and

${\displaystyle f'_{21}=1+{\frac {1}{9}}x^{2}t_{1}t_{2}-{\frac {1}{9}}x^{2}t_{1}^{2}-{\frac {13}{135}}t_{1}^{2}x^{4}+{\frac {13}{135}}t_{1}t_{2}x^{4}+{\frac {2}{135}}t_{1}^{4}x^{2}+{\frac {2}{45}}t_{1}^{2}t_{2}^{2}x^{2}-{\frac {8}{135}}t_{1}^{3}t_{2}x^{2}}$

${\displaystyle -{\frac {1147t_{1}^{2}x^{6}}{14175}}+{\frac {1147t_{1}t_{2}x^{6}}{14175}}+{\frac {13}{525}}t_{1}^{4}x^{4}+{\frac {878t_{1}^{2}t_{2}^{2}x^{4}}{14175}}-{\frac {1229t_{1}^{3}t_{2}x^{4}}{14175}}-{\frac {1}{525}}t_{1}^{6}x^{2}+{\frac {2}{105}}t_{1}^{3}t_{2}^{3}x^{2}-{\frac {1}{35}}t_{1}^{4}t_{2}^{2}x^{2}+{\frac {2}{175}}t_{1}^{5}t_{2}x^{2}}$
${\displaystyle -{\frac {2939t_{1}^{2}x^{8}}{42525}}+{\frac {2939t_{1}t_{2}x^{8}}{42525}}+{\frac {1327t_{1}^{4}x^{6}}{42525}}+{\frac {2896t_{1}^{2}t_{2}^{2}x^{6}}{42525}}-{\frac {4223t_{1}^{3}t_{2}x^{6}}{42525}}}$
${\displaystyle -{\frac {199t_{1}^{6}x^{4}}{42525}}+{\frac {20}{567}}t_{1}^{3}t_{2}^{3}x^{4}-{\frac {97t_{1}^{4}t_{2}^{2}x^{4}}{1701}}+{\frac {1124t_{1}^{5}t_{2}x^{4}}{42525}}+{\frac {2t_{1}^{8}x^{2}}{8505}}+{\frac {2}{243}}t_{1}^{4}t_{2}^{4}x^{2}-{\frac {16t_{1}^{5}t_{2}^{3}x^{2}}{1215}}+{\frac {8t_{1}^{6}t_{2}^{2}x^{2}}{1215}}-{\frac {16t_{1}^{7}t_{2}x^{2}}{8505}}}$
${\displaystyle -{\frac {2953639t_{1}^{2}x^{10}}{49116375}}+{\frac {2953639t_{1}t_{2}x^{10}}{49116375}}+{\frac {1740446t_{1}^{4}x^{8}}{49116375}}+{\frac {3411068t_{1}^{2}t_{2}^{2}x^{8}}{49116375}}-{\frac {5151514t_{1}^{3}t_{2}x^{8}}{49116375}}}$
${\displaystyle -{\frac {382048t_{1}^{6}x^{6}}{49116375}}+{\frac {152648t_{1}^{3}t_{2}^{3}x^{6}}{3274425}}-{\frac {87874t_{1}^{4}t_{2}^{2}x^{6}}{1091475}}+{\frac {2046658t_{1}^{5}t_{2}x^{6}}{49116375}}}$
${\displaystyle +{\frac {7472t_{1}^{8}x^{4}}{9823275}}+{\frac {25931t_{1}^{4}t_{2}^{4}x^{4}}{1403325}}-{\frac {45641t_{1}^{5}t_{2}^{3}x^{4}}{1403325}}+{\frac {5377t_{1}^{6}t_{2}^{2}x^{4}}{280665}}-{\frac {57697t_{1}^{7}t_{2}x^{4}}{9823275}}}$
${\displaystyle -{\frac {1382t_{1}^{10}x^{2}}{49116375}}+{\frac {2764t_{1}^{5}t_{2}^{5}x^{2}}{779625}}-{\frac {2764t_{1}^{6}t_{2}^{4}x^{2}}{467775}}+{\frac {11056t_{1}^{7}t_{2}^{3}x^{2}}{3274425}}-{\frac {1382t_{1}^{8}t_{2}^{2}x^{2}}{1091475}}+{\frac {2764t_{1}^{9}t_{2}x^{2}}{9823275}}}$?

(These Taylor expansions are also available within the mathematica notebook HOMFLY Braidor - Braidor Computations.nb).