spinsl2-ii.workingwiki.tex
\documentclass[12pt]{amsart}

\usepackage{amsmath}%
\usepackage{amsfonts}%
\usepackage{amssymb}%
\usepackage{graphicx}
\usepackage[all]{xy}
\usepackage{array}

%--------------------------------------------------------
\usepackage{tikz}
\usetikzlibrary{decorations.shapes}
\usetikzlibrary{decorations.text}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{shapes}
\usetikzlibrary{shapes.symbols}
\input{tikzstyledefs}
\tikzstyle{every picture}=[semithick,baseline=0pt,heightone,label distance=-2mm]
%--------------------------------------------------------

% Theorem like environments

\newtheorem{theorem}{Theorem}
\theoremstyle{plain}
\newtheorem{acknowledgement}{Acknowledgement}
\newtheorem{algorithm}{Algorithm}
\newtheorem{axiom}{Axiom}
\newtheorem{case}{Case}
\newtheorem{claim}{Claim}
\newtheorem{conclusion}{Conclusion}
\newtheorem{condition}{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}{Lemma}
\newtheorem{notation}{Notation}
\newtheorem{problem}{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{solution}{Solution}
\newtheorem{summary}{Summary}
\numberwithin{equation}{section}
\newtheorem{fact}{Fact}
\newtheorem{observation}{Observation}
\newtheorem{comment}{Comment}

\numberwithin{theorem}{section}
%\numberwithin{figure}{section}
\numberwithin{equation}{section}

\def\C{\mathbb{C}}
\def\na{\underline{a}}
\def\nb{\underline{b}}
\def\nc{\underline{c}}
\def\pa{\overline{a}}
\def\pb{\overline{b}}
\def\pc{\overline{c}}
\def\tr{\mathrm{tr}}
\def\hTheta{\hat{\Theta}}
\def\sixj#1#2#3#4#5#6{\begin{bmatrix}#1 & #2 & #6\\#4 & #3 & #5\end{bmatrix}}
\def\sixjt#1#2#3#4#5#6{\bigl[\begin{smallmatrix}#1 & #2 & #6\\#4 & #3 & #5\end{smallmatrix}\bigr]}
\def\splus#1{\overline{#1}}
\def\sminus#1{\underline{#1}}
\def\edges#1#2#3{\mathfrak{e}_{#1}(#2,#3)}
\def\signs#1#2#3{\mathfrak{s}_{#1}(#2,#3)}
\def\mfe{\mathfrak{e}}
\def\mfb{\mathfrak{b}}
\def\mff{\mathfrak{f}}
\def\mfs{\mathfrak{s}}
\def\adm#1#2#3{#1\in\lceil #2,#3\rfloor}
\def\Fus#1#2#3#4#5{\mathfrak{F}_{#1}\left(\begin{smallmatrix}#3 & #5\\ #2 & #4\end{smallmatrix}\right)}
\def\nFus#1#2#3#4#5{\hat{\mathfrak{F}}_{#1}\left(\begin{smallmatrix}#3 & #5\\ #2 & #4\end{smallmatrix}\right)}

%--------------Sean's commands-----------------------------
\newcommand{\Tr}[1]{\mathrm{tr}(#1)}
\newcommand{\xb}{\mathbf{X}}
\newcommand{\yb}{\mathbf{Y}}
\newcommand{\zb}{\mathbf{Z}}
\newcommand{\pol}{\mathrm{pol}}
\newcommand{\ub}{\mathbf{U}}
\newcommand{\vb}{\mathbf{V}}
\newcommand{\wb}{\mathbf{W}}
\newcommand{\dg}[1]{|\!|#1|\!|}
\newcommand{\tdg}[1]{|\!|#1|\!|_{\mathrm{tr}}}
\newcommand{\aq}{/\!\!/}
\newcommand{\X}{\mathfrak{X}}
\newcommand{\Y}{\mathfrak{Y}}
\newcommand{\R}{\mathfrak{R}}
\newcommand{\hm}{\mathrm{Hom}}
\newcommand{\SL}{\mathrm{SL}(2,\C)}
\newcommand{\SLm}[1]{\mathrm{SL}(#1,\C)}
\newcommand{\ot}{\mathrm{Out}}
\newcommand{\F}{\mathtt{F}}
\newcommand{\G}{\mathfrak{G}}
\newcommand{\xt}{\mathtt{x}}
\newcommand{\yt}{\mathtt{y}}
\newcommand{\wt}{\mathtt{w}}
\newcommand{\bt}{\mathtt{b}}
\newcommand{\rt}{\mathtt{r}}
\newcommand{\M}{\mathbf{M}}
\newcommand{\id}{\mathbf{I}}
\newcommand{\D}{\mathfrak{D}}
\newcommand{\ti}[1]{t_{(#1)}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\iadm}[1]{\lceil #1 \rfloor}
\newcommand{\vi}{\vec{i}}
\newcommand{\vj}{\vec{j}}
\newcommand{\vk}{\vec{k}}
%-------------------------------------------------------

%---------------------Elisha's Old Commands--------------
\newcommand{\tmx}[2]{\left(\begin{matrix}#1\\#2\end{matrix}\right)}                      % gives a two-row matrix
\newcommand{\tmxt}[2]{\bigl(\begin{smallmatrix}#1\\#2\end{smallmatrix}\bigr)} % gives a two-row matrix in the text
\newcommand{\Adm}[3]{#3\in\lceil #1,#2\rfloor}
\newcommand{\N}{\mathbb{N}}
\newcommand{\ch}[3]{\chxx_{\sss #2,#3}^{\sss #1}}
\newcommand{\chh}[3]{\chxx^{\sss #2,#3}_{\sss #1}}
\def\sss{\scriptscriptstyle}
\newcommand{\chxx}{\raisebox{2pt}{$\chi$}}
\newcommand{\cspan}{\C\,}
\def\pfrac#1#2{\left(\frac{#1}{#2}\right)}
\newcommand{\bs}[3]{{#1}^*_{#2}\otimes {#1}_{#3}}
\def\vprod#1{V^{\otimes #1}}
\def\vprods#1{(V^*)^{\otimes #1}}
\newcommand{\imx}[1]{\tmx{#1_{11} & #1_{12}}{#1_{21} & #1_{22}}}
\newcommand{\cgright}{\C[\SL]_{\sss\textup{right}}}
%-------------------------------------------------------

%--------------------------------------------------------
\begin{document}
\title[Computing Central Functions]{Computing $\mathrm{SL}(2,\C)$ Central Functions with Spin Networks}
\author{Sean Lawton and Elisha Peterson}
\address[Sean Lawton]{IST}
\email[]{slawton@math.ist.utl.pt}
\address[Elisha Peterson]{United States Military Academy}
\email[]{elisha.peterson@usma.edu}
\date{\today}

\begin{abstract}
In this paper we generalize our notion of a central function to arbitrary rank free groups and then explore the ring structure for the rank 3 case.
\end{abstract}
\maketitle

\tableofcontents

\section{Introduction}\label{intro-section}\label{s:introduction}
\input{intro.workingwiki.tex}

\section{Spin Networks and Trace Diagrams}\label{s:spinnets}
\input{spinnetsintro.tex}

\section{Rank $r$ Central Functions}\label{s:centralfunctions}
\input{rankrcentral.workingwiki.tex}

\section{Trace Diagram Recurrences}\label{s:recurrences}
\input{tracediagramreview.tex}
\input{simplerecurrence.tex}
\input{nonsimplerecurrence.tex}

\section{Rank Three Central Functions}\label{s:rankthree}
\input{rank3.workingwiki.tex}

\section{Conclusion}
%TODO work in some consequences of the recurrence formulas

\section*{Acknowledgments}
We would like to thank Bill Goldman for introducing us to this topic.  A very special thanks is due to Suhyoung Choi and to Korea Advanced Institute of Science and Technology (KAIST) for generously inviting us to speak for their seventh Geometric Topology Fair.  This paper comes out of our time at KAIST.

\bibliographystyle{amsalpha}
\bibliography{bib}
%\input{spinsl2-ii.workingwiki.bbl}

\end{document}

Return

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License