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\begin{document}
\parindent 0in

%\parbox[b]{4in}{
%  {\tiny Dror Bar-Natan: MAT 1350 Knot Theory: \LARGE\bf Tentative Plan}
%  }
\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}: \href{http://drorbn.net/21-1350}{MAT 1350 Knot Theory}:
}~{\LARGE\bf Homework Assignment 5}
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due December 8 at midnight
}~%
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/21-1350}
}~\parbox[b]{0.3in}{\tiny\href{http://www.math.toronto.edu/~drorbn/bv.html}{Better\newline vaxxed!}}

\vskip -3mm
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\vskip 5mm

\begin{center}\begin{minipage}{6in}

{\bf Problem 1.} By whatever means you wish, compute $\sum_D\frac{1}{|\Aut(D)|}$ where $D$ runs over all {\red connected} diagrams with exactly $m=10$ quadrivalent vertices (vertices that connect to 4 edges). For example, for $m=1$ this number is $\frac{1}{2^3}=\frac{1}{8}$, for $m=2$ it is $\frac{1}{2\cdot 4!}+\frac{1}{2^4}=\frac{1}{12}$ and for $m=3$ it is $\frac{11}{96}$.

\vskip 5mm{\bf Problem 2.} By whatever means you wish, find the generating function for the multiplication map $m^{ij}_k\colon A_i\otimes A_j\to A_k$ of the associative algebra $A\coloneqq\langle \ba,\bx\rangle/([\ba,\bx]=\bx)$, relative to the ordered basis $(\ba,\bx)$.

\vskip 5mm{\bf Problem 3.} By whatever means you wish, yet without using results derived in class for $QU$, prove that the element $R_{ij}\coloneqq\bbe^{\hbar \bb_i\ba_j+\hbar \by_i\bx_j}$ of $(\calU(sl_{2+}^0)_i\otimes\calU(sl_{2+}^0)_j)\llbracket\hbar\rrbracket$, with $sl_{2+}^0$ being $sl_{2+}^\epsilon$ at $\epsilon=0$, satisfies the Yang-Baxter equation (Reidemeister 3) in $\calU(sl_{2+}^0)^{\otimes 3}\llbracket\hbar\rrbracket$.

\end{minipage}\end{center}

\end{document}


