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

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

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

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

Examples.

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

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

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

The Equations

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

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

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

1. $[t_{i}^{k},t_{j}]=x\sigma _{ij}(t_{i}^{k}-t_{j}^{k})$ and therefore $[e^{t_{i}},t_{j}]=x\sigma _{ij}(e^{t_{i}}-e^{t_{j}})$.
2. $\Delta (t_{i}^{k})=t_{i+1}^{k}+{\frac {x}{2}}\left({\frac {(t_{i+1}+x)^{k}-t_{1}^{k}}{t_{i+1}+x-t_{1}}}-{\frac {(t_{i+1}-x)^{k}-t_{1}^{k}}{t_{i+1}-x-t_{1}}}\right)+\sigma _{1,i+1}{\frac {x}{2}}\left({\frac {(t_{i+1}+x)^{k}-t_{1}^{k}}{t_{i+1}+x-t_{1}}}+{\frac {(t_{i+1}-x)^{k}-t_{1}^{k}}{t_{i+1}-x-t_{1}}}\right)$ and therefore $\Delta (e^{t_{i}})=e^{t_{i+1}}+{\frac {x}{2}}\left({\frac {e^{t_{i+1}+x}-e^{t_{1}}}{t_{i+1}+x-t_{1}}}-{\frac {e^{t_{i+1}-x}-e^{t_{1}}}{t_{i+1}-x-t_{1}}}\right)+\sigma _{1,i+1}{\frac {x}{2}}\left({\frac {e^{t_{i+1}+x}-e^{t_{1}}}{t_{i+1}+x-t_{1}}}+{\frac {e^{t_{i+1}-x}-e^{t_{1}}}{t_{i+1}-x-t_{1}}}\right)$. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables $x$, $t_{1}$ and $t_{i+1}$, and then the "true" $x$, $t_{1}$ and $t_{i+1}$ 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.

A Numerology Problem

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

$f_{12}=x+{\frac {xt_{2}}{3}}-{\frac {xt_{1}}{3}}$

$-{\frac {1}{5}}t_{1}x^{3}+{\frac {t_{2}x^{3}}{5}}+{\frac {t_{1}^{3}x}{45}}-{\frac {t_{2}^{3}x}{45}}+{\frac {1}{15}}t_{1}t_{2}^{2}x-{\frac {1}{15}}t_{1}^{2}t_{2}x$

$-{\frac {1}{7}}t_{1}x^{5}+{\frac {t_{2}x^{5}}{7}}+{\frac {11}{315}}t_{1}^{3}x^{3}-{\frac {11}{315}}t_{2}^{3}x^{3}+{\frac {11}{105}}t_{1}t_{2}^{2}x^{3}-{\frac {11}{105}}t_{1}^{2}t_{2}x^{3}$

$-{\frac {2t_{1}^{5}x}{945}}+{\frac {2t_{2}^{5}x}{945}}-{\frac {2}{189}}t_{1}t_{2}^{4}x+{\frac {4}{189}}t_{1}^{2}t_{2}^{3}x-{\frac {4}{189}}t_{1}^{3}t_{2}^{2}x+{\frac {2}{189}}t_{1}^{4}t_{2}x$

$-{\frac {1}{9}}t_{1}x^{7}+{\frac {t_{2}x^{7}}{9}}+{\frac {598t_{1}^{3}x^{5}}{14175}}-{\frac {598t_{2}^{3}x^{5}}{14175}}+{\frac {1619t_{1}t_{2}^{2}x^{5}}{14175}}-{\frac {1619t_{1}^{2}t_{2}x^{5}}{14175}}$

$-{\frac {74t_{1}^{5}x^{3}}{14175}}+{\frac {74t_{2}^{5}x^{3}}{14175}}-{\frac {74t_{1}t_{2}^{4}x^{3}}{2835}}+{\frac {148t_{1}^{2}t_{2}^{3}x^{3}}{2835}}-{\frac {148t_{1}^{3}t_{2}^{2}x^{3}}{2835}}+{\frac {74t_{1}^{4}t_{2}x^{3}}{2835}}$

$+{\frac {t_{1}^{7}x}{4725}}-{\frac {t_{2}^{7}x}{4725}}+{\frac {1}{675}}t_{1}t_{2}^{6}x-{\frac {1}{225}}t_{1}^{2}t_{2}^{5}x+{\frac {1}{135}}t_{1}^{3}t_{2}^{4}x-{\frac {1}{135}}t_{1}^{4}t_{2}^{3}x+{\frac {1}{225}}t_{1}^{5}t_{2}^{2}x-{\frac {1}{675}}t_{1}^{6}t_{2}x$

$-{\frac {1}{11}}t_{1}x^{9}+{\frac {t_{2}x^{9}}{11}}+{\frac {2414t_{1}^{3}x^{7}}{51975}}-{\frac {2414t_{2}^{3}x^{7}}{51975}}+{\frac {53243t_{1}t_{2}^{2}x^{7}}{467775}}-{\frac {53243t_{1}^{2}t_{2}x^{7}}{467775}}$

$-{\frac {4058t_{1}^{5}x^{5}}{467775}}+{\frac {4058t_{2}^{5}x^{5}}{467775}}-{\frac {3904t_{1}t_{2}^{4}x^{5}}{93555}}+{\frac {782t_{1}^{2}t_{2}^{3}x^{5}}{10395}}-{\frac {782t_{1}^{3}t_{2}^{2}x^{5}}{10395}}+{\frac {3904t_{1}^{4}t_{2}x^{5}}{93555}}$

$+{\frac {331t_{1}^{7}x^{3}}{467775}}-{\frac {331t_{2}^{7}x^{3}}{467775}}+{\frac {331t_{1}t_{2}^{6}x^{3}}{66825}}-{\frac {331t_{1}^{2}t_{2}^{5}x^{3}}{22275}}+{\frac {331t_{1}^{3}t_{2}^{4}x^{3}}{13365}}-{\frac {331t_{1}^{4}t_{2}^{3}x^{3}}{13365}}+{\frac {331t_{1}^{5}t_{2}^{2}x^{3}}{22275}}-{\frac {331t_{1}^{6}t_{2}x^{3}}{66825}}$

$-{\frac {2t_{1}^{9}x}{93555}}+{\frac {2t_{2}^{9}x}{93555}}-{\frac {2t_{1}t_{2}^{8}x}{10395}}+{\frac {8t_{1}^{2}t_{2}^{7}x}{10395}}-{\frac {8t_{1}^{3}t_{2}^{6}x}{4455}}+{\frac {4t_{1}^{4}t_{2}^{5}x}{1485}}-{\frac {4t_{1}^{5}t_{2}^{4}x}{1485}}+{\frac {8t_{1}^{6}t_{2}^{3}x}{4455}}-{\frac {8t_{1}^{7}t_{2}^{2}x}{10395}}+{\frac {2t_{1}^{8}t_{2}x}{10395}}$

and

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

$-{\frac {1147t_{1}^{2}x^{6}}{14175}}+{\frac {1147t_{1}t_{2}x^{6}}{14175}}+{\frac {13}{525}}t_{1}^{4}x^{4}+{\frac {878t_{1}^{2}t_{2}^{2}x^{4}}{14175}}-{\frac {1229t_{1}^{3}t_{2}x^{4}}{14175}}-{\frac {1}{525}}t_{1}^{6}x^{2}+{\frac {2}{105}}t_{1}^{3}t_{2}^{3}x^{2}-{\frac {1}{35}}t_{1}^{4}t_{2}^{2}x^{2}+{\frac {2}{175}}t_{1}^{5}t_{2}x^{2}$

$-{\frac {2939t_{1}^{2}x^{8}}{42525}}+{\frac {2939t_{1}t_{2}x^{8}}{42525}}+{\frac {1327t_{1}^{4}x^{6}}{42525}}+{\frac {2896t_{1}^{2}t_{2}^{2}x^{6}}{42525}}-{\frac {4223t_{1}^{3}t_{2}x^{6}}{42525}}$

$-{\frac {199t_{1}^{6}x^{4}}{42525}}+{\frac {20}{567}}t_{1}^{3}t_{2}^{3}x^{4}-{\frac {97t_{1}^{4}t_{2}^{2}x^{4}}{1701}}+{\frac {1124t_{1}^{5}t_{2}x^{4}}{42525}}+{\frac {2t_{1}^{8}x^{2}}{8505}}+{\frac {2}{243}}t_{1}^{4}t_{2}^{4}x^{2}-{\frac {16t_{1}^{5}t_{2}^{3}x^{2}}{1215}}+{\frac {8t_{1}^{6}t_{2}^{2}x^{2}}{1215}}-{\frac {16t_{1}^{7}t_{2}x^{2}}{8505}}$?