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\def\bbQ{{\mathbb Q}}
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\def\calL{{\mathcal L}}
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\def\FL{\text{\it FL}}
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\begin{document}
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{\LARGE\bf A Polynomial Time Knot Polynomial}\hfill
\parbox[b]{3in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/2015-07/PolyPoly}
  \newline\null\hfill initiated 17/6/15;
  modified \today, \ampmtime.
}

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\begin{multicols}{2}\raggedcolumns

{\bf Deriving Gassner.} $\yellowm{\calL^{2Dw}}$ is $\bbQ\llbracket\yellowm{b_i}\rrbracket\langle\yellowm{a_{ij}}\rangle$ modulo locality,
$[a_{ij},a_{ik}] = 0$,
$[a_{ik},a_{jk}] = -[a_{ij},a_{jk}] = b_ja_{ik}-b_ia_{jk}$,
and (mod~$\langle a_{ii}\rangle$) $[a_{ij},a_{ji}] = b_ia_{ji}-b_ja_{ij}$.
Acts on $\yellowm{V} = {\bbQ\llbracket b_i\rrbracket\langle\yellowm{x_i}=a_{i\infty}\rangle}$ by $[a_{ij},x_i]=0$, $[a_{ij},x_j] = b_ix_j-b_jx_i$. Hence $e^{\ad a_{ij}}x_i=x_i$, $e^{\ad a_{ij}}x_j = e^{b_i}x_j+\frac{b_j}{b_i}(1-e^{b_i})x_i$. Renaming $\yellowm{y_i}=x_i/b_i$, $\yellowm{t_i}=e^{b_i}$, get $[e^{\ad a_{ij}}]_{y_i,y_j} = \begin{pmatrix} 1 & 1-t_i \\ 0 & t_i \end{pmatrix}$.

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