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% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
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\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real nubers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\st}{ \ : \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\img}{\mathrm{im}} % image
\newcommand{\into}{\hookrightarrow} % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
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% SOME ENVIRONMENTS

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    }

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\title{MAT180 HW04}
\author{(ADD NAME)}
\date{Due 4/29/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle


{\tiny
\paragraph{Reminder} 
Your homework submission must be typed up in full sentences, with proper mathematical formatting. The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{https://detexify.kirelabs.org/classify.html}
\end{itemize}
}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

Let $H$ denote the following diagram of the (unoriented) Hopf link: 
\begin{center}
\begin{tikzpicture}[scale=.5]
    \draw (1,0) arc (-30:300:2cm);
    \begin{scope}[xscale=-1,yscale=-1, yshift=-2cm, xshift=-1.3cm]
        \draw (1,0) arc (-30:300:2cm);
    \end{scope}
\end{tikzpicture}
\end{center}

\bea

\item Draw the \emph{cube of resolutions} for the diagram above. That is, replace the binary strings below with the associated smoothings of the diagram:
\begin{center}
\begin{tikzpicture}
    \node (00) at (0,0) {00};
    \node (10) at (2,1) {10};
    \node (01) at (2,-1) {01};
    \node (11) at (4,0) {11};
    \draw (00) -- (10) -- (11);
    \draw (00) -- (01) -- (11);
\end{tikzpicture}
\end{center}
\begin{remark}
The cube of resolutions can be viewed as a graph representing the partially ordered set $\{0,1\}^n$ where the bit string $a = a_1a_2\ldots a_n \leq b_1b_2\ldots b_n$ iff for each $i$, $a_i \leq b_i$. 
\end{remark}

\item Compute the Kauffman bracket polynomial $\langle H \rangle$. 

\item Use your computation of $\langle H \rangle$ to compute the Jones polynomials of the positively linked Hopf link $H^+$ and the negatively linked $H^-$ in terms of the variable $q$.


\ee







\end{document}
