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\newtheorem{thm}{Theorem}

\newtheorem{project}{Project Problem}




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\newcommand{\R}{\mathbb{R}}
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\definecolor{Maroon}{rgb}{.5,.1,.25}% Used for Trinity University


\title[Knot Theory]{A brief Incursion into Knot Theory}
%\title[Short Title]{Long Title}
%\subtitle{with a detour to Geometric Analysis}


%\author[Name to appear in every slide]{Author name to appear in Title}
\author[]{Eduardo Balreira}


\institute[Balreira - Trinity University]{%
   \includegraphics[width=1.0cm]{ltr-logo}\\%remove this line if no logo.
   \textcolor{Maroon}{{\Large Trinity University}}\\
   Mathematics Department
}

\date[Major Seminar] % (optional, should be abbreviation of conference name)
{Major Seminar, Fall 2008}
% - Either use conference name or its abbreviation.



\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\section{Outline}
%\frame{\tableofcontents}



\begin{frame}
  \frametitle{Outline}
  \tableofcontents[pausesections]
  % You might wish to add the option [pausesections] to display one at a time
\end{frame}
%
%

\section{A Fundamental Problem}

\begin{frame}
 \frametitle{A Fundamental Problem}

\begin{center}
Given two objects, how to tell them apart?
\end{center}

\vfill


   \begin{itemize}

    \item When are two surfaces homeomorphic?

    \begin{center}
     \includegraphics{spheretorus.eps}
    \end{center}


        \pause

        \vfill

    \item When are two knots equivalent?

    \begin{center}
    \includegraphics{unknotvstrefoil1.eps}
    \end{center}


   \end{itemize}
\end{frame}

\begin{frame}
 \frametitle{A Fundamental Problem}
  \framesubtitle{Main Research Area}

  Injectivity via Geometric and Topological Methods
  
  \vfill

   \begin{itemize}

    \item Foliation Theory

        \pause

        \vfill

    \item Spectral Theory

        \pause

        \vfill

    \item Variational Calculus

        \pause

        \vfill

    \item Global Embedding of Submanifolds

   \end{itemize}
\end{frame}


\begin{frame}
 \frametitle{Sample Topics}

\uncover<2->{\begin{project}[Foliation Theory] Discuss Fibrations versus Foliations, e.g., characterize the foliations of the Euclidean Plane, that is, $\R^2$.
\end{project}}

\vfill

\uncover<3->{\begin{project}[Spectral Theory] Discuss the eigenvalues of the Laplacian on a bounded region of $\R^n$, e.g., Rayleigh and Min-Max Methods.
\end{project}}

\end{frame}


\begin{frame}
 \frametitle{Sample Topics}


\uncover<2->{\begin{project}[Variational Calculus] Discuss any result in the Geometric Analysis report, e.g., understand the MPT and its proof.
\end{project}}

\vfill

\uncover<3->{\begin{project}[Geometry] Learn about Classification of Surfaces via the Euler Characteristic and/or understand the Gauss-Bonnet formula
\[
\int_M K~dA = 2\pi\chi(M)
\]
\end{project}}



\end{frame}


\section{Knot Theory}

\begin{frame}
 \frametitle{Knot Theory}
 \framesubtitle{When can tell the difference between Knots?}

Are the knots below the same?

 \begin{center}
  \includegraphics{threetypes.eps}
\end{center}

\end{frame}

\begin{frame}
 \frametitle{Knot Theory}
 \framesubtitle{When can tell the difference between Knots?}

How about these?

 \begin{center}
  \includegraphics{twotypes.eps}
\end{center}

\end{frame}

\begin{frame}
 \frametitle{Knot Theory}
 \framesubtitle{Some Formalism}

A knot is an injective map $h:S^1\to \R^3$

\vfill

   \begin{itemize}

    \item Picture in the plane (or slide) - Diagram with crossing

        \pause

        \vfill


    \item Tame Knots
    \pause
     \begin{itemize}

    \item Finite Number of arcs

        \pause

        \vfill


    \item Only two strands at a crossing

        \pause

        \vfill


    \item ``nice''
    \pause

     \end{itemize}


    \item Invariant Property: $K_1 \sim K_2$ if,

        \pause

        \vfill


    There is a homeomorphism $\fee:\R^3\to\R^3$ such that
    \[\fee(K_1) = K_2\]

   \end{itemize}

\end{frame}

\begin{frame}
 \frametitle{Knot Theory}
 \framesubtitle{Isotopy Problem}

\begin{center}
The map $\fee$ itself must be ``nice'' $\longleftrightarrow$ \textbf{Isotopic to the Identity}
\end{center}


\vfill

\begin{itemize}

\item No situations such as:

\begin{center}
  \includegraphics{notisotopy.eps}
\end{center}

\end{itemize}

\end{frame}


\begin{frame}
 \frametitle{Knot Theory - Diagrams}
\only<1>{
\begin{center}
Two Knots with \textbf{similar} Diagram must be the same.
\end{center}

\begin{center}
  \includegraphics{braid/braid1.eps}
\end{center}
}

\only<2>{
\begin{center}
Two Knots with \textbf{similar} Diagram must be the same.
\end{center}

\begin{center}
  \includegraphics{braid/braid2.eps}
\end{center}
}
\only<3>{
\begin{center}
Two Knots with \textbf{similar} Diagram must be the same.
\end{center}

\begin{center}
  \includegraphics{braid/braid3.eps}
\end{center}
}
\only<4>{
\begin{center}
Two Knots with \textbf{similar} Diagram must be the same.
\end{center}

\begin{center}
  \includegraphics{braid/braid4.eps}
\end{center}
}
\only<5>{
\begin{center}
Two Knots with \textbf{similar} Diagram must be the same.
\end{center}

\begin{center}
  \includegraphics{braid/braid5.eps}
\end{center}
}


\end{frame}


\begin{frame}
 \frametitle{Knot Theory - Diagrams}


\begin{center}
Even if not exactly the diagram same, they could be the same...
\end{center}


 \begin{center}
  \includegraphics{twotypes.eps}
\end{center}


\end{frame}


\section{Reidemeister Moves}

\begin{frame}
 \frametitle{Moves that preserve the Knot}

\begin{center}
{\Large\textbf{Reidemeister Moves}}
\end{center}

\jl

     \begin{itemize}

    \item Type I: Put or Take out a kink.

    \pause

    \vfill

    \item Type II: Slide a strand over/under to creat/remove two crossings.

    \pause

    \vfill

    \item Type III: Slide a strand across a crossing.

     \end{itemize}


\end{frame}

\begin{frame}
 \frametitle{Type I}
  \framesubtitle{Put or Take out a kink.}

 \begin{center}
  \includegraphics{typeI.eps}
\end{center}

\end{frame}



\begin{frame}
 \frametitle{Type II}
  \framesubtitle{Slide a strand over/under to creat/remove two crossings.}

 \begin{center}
  \includegraphics{typeII.eps}
\end{center}

\end{frame}

\begin{frame}
 \frametitle{Type III}
  \framesubtitle{Slide a strand across a crossing}

 \begin{center}
  \includegraphics{typeIII.eps}
\end{center}

\end{frame}

\begin{frame}
 \frametitle{Unknotting a Knot}
  \framesubtitle{Using Reidemeister moves}

\only<1>{
    \centerline{\includegraphics{Reid1.eps}}
  }
  \only<2>{
    \centerline{\includegraphics{Reid2.eps}}
  }
  \only<3>{
    \centerline{\includegraphics{Reid3.eps}}
  }
  \only<4>{
    \centerline{\includegraphics{Reid4.eps}}
  }

\end{frame}



\begin{frame}
 \frametitle{Knot Theory}
  \framesubtitle{A Possible classification?}

\uncover<2->{\begin{thm}[Reidemeister, 1932]
Two Knot diagrams of the same knot can be deformed into each other by a finite number of moves of Type I, II, and III.
\end{thm}}

\vfill

\uncover<3->{\begin{project}
Discuss the result above, its proof, and its significance in other areas. For instance, Physics - the structure of the atom and Chemistry - the structure of the DNA replication.
\end{project}}

\end{frame}




\section{Invariants}

\begin{frame}
\frametitle{Invariants}


 \begin{itemize}
  \item Colorability

  \pause

  \vfill

  \item Unknotting number

  \pause

  \vfill

  \item Genus of a knot

  \pause

  \vfill

  \item Knot Group

  \pause

  \vfill

  \item Polynomial Invariants

 \end{itemize}

\end{frame}


\begin{frame}
 \frametitle{Sample Problems in Knot Theory}

\uncover<2->{\begin{project}[Colorability]
Understand the definition and application as well as generalizations for $\mod p$ colorability. This project has a Topological and Number Theoretical flavor.
\end{project}}


\vfill

\uncover<3->{\begin{project}[Unknotting number] Discuss classical result and classification of knots via crossing and unknotting numbers, e.g., minimal diagrams and prime knots.
\end{project}}


\end{frame}

\begin{frame}
 \frametitle{More Sample Problems in Knot Theory}


\uncover<2->{\begin{project}[Genus of a knot]
Learn surface theory in order to understand Seifert surfaces. Show that the genus of a knot, $\dfrac{-\chi(M)+1}{2}$ is a knot invariant.
\end{project}}

\vfill


\uncover<3->{ \begin{center}
  \includegraphics{seifert.eps}
\end{center}}


\end{frame}


\begin{frame}
 \frametitle{Even More Sample Problems in Knot Theory}


\uncover<1->{\begin{project}[Knot Group]
Learn to compute the knot group and discuss group presentations and possibly learn about representation theory.
\end{project}}

\vfill

\uncover<2->{
\[
\pi_1 (\R^3-T) = \langle a,b |~aba = bab\rangle = \langle x,y |~x^2 = y^3\rangle
\]

\begin{center}
Can we make sense of the expressions above?
\end{center}

}


\end{frame}

\begin{frame}
 \frametitle{One More Sample Problem in Knot Theory}


\begin{project}[Polynomial Invariants]
Understand and learn to compute a polynomial invariant of a knot, e.g.,
Bracket, Kauffman, Jones, Homfly, and Alexander Polynomials.
\end{project}

\vfill

\begin{itemize}

\item All rules come from the topology of the knots (i.e., crossing types)
\pause
\vfill

\item Some are formal Algebraic Computations.
\pause
\vfill
\item Others are Geometric in nature.

\end{itemize}


\end{frame}


\subsection{Colorability}

\begin{frame}
  \frametitle{Invariants}
  \framesubtitle{Colorability}
\begin{center}
  $K$ is colorable $\leftrightarrow$ each arc has one of 3 colors
\end{center}

  \begin{itemize}
  \item<1-> At least two of the colors are used
  \item<1-> At crossing, either all different or all the same color.
  \end{itemize}
    \only<1>{\centerline{%
      \includegraphics{tricolor0.eps}
    }}
  \only<2>{%
    \centerline{%
      \includegraphics{tricolor1.eps}
    }
  }
  \only<3->{%
    \centerline{%
      \includegraphics{tricolor2.eps}
    }
  }
\end{frame}

\begin{frame}
 \frametitle{Colorability}
  \framesubtitle{The Unknot and Trefoil are different!}

    \centerline{%
      \includegraphics{unknotvstrefoil.eps}
    }
 \end{frame}


\subsection{The Knot Group}

\begin{frame}
 \frametitle{An Algebraic Invariant}

Topologically, every knot is equivalent to $S^1$. DONE?

\vfill

 \begin{itemize}
  \item<2-> Knots are different via their embedding in $\R^3$.

  \vfill

  \item<3-> Better question: $K_1\sim K_2$ if $\exists~\fee:\R^3\to\R^3$ with
\[ \fee(K_1) = K_2\]

\vfill

  \item<4-> Consider $\fee_{\left|_{\R^3-K_1}\right.}: \R^3-K_1 \to \R^3-K_2$

  \vfill

  \item<5-> Thus \[\pi_1(\R^3-K_1) \simeq \pi_1(\R^3-K_2)\]

\vfill

  \item<6-> \noi Same \textbf{Fundamental Group}

  \end{itemize}

\end{frame}

\begin{frame}
 \frametitle{The Knot Group}
  \framesubtitle{An Algebraic Invariant}

 \begin{itemize}
  \item $\pi_1(M)$ is the Fundamental Group of $M$

  \pause
  \vfill

  \item For a knot $K$, $\pi_1(\R^3-K_1)$ is the \textbf{Knot Group} of $K$

  \pause
  \vfill

  \item MATH 4365 - Topology (New Course! Fall 09)

  \pause
  \vfill

  \item Some Algebra background would be very nice...

  \end{itemize}

\end{frame}


\begin{frame}
 \frametitle{An Algebraic Invariant}
 \framesubtitle{Types of Crossings}

\begin{figure}[h]
\begin{center}
\includegraphics{Crossingtypes.eps}
\end{center}
\end{figure}



\end{frame}



\begin{frame}
 \frametitle{Computing the Knot group}

\begin{itemize}

\item Label each arc with a letter.

\pause

\item At a crossing: the emerging arc labeled by the conjugate as follows:

\begin{figure}[h]
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{1}{$c=bab^{-1}$}\psfrag{2}{$c=b^{-1}ab$}
\includegraphics{conjugate.eps}
\end{psfrags}
\end{center}
\end{figure}

\end{itemize}

\end{frame}


\begin{frame}
 \frametitle{Computing the Knot group}
  \framesubtitle{The Trefoil Group}
%
%
%%\begin{figure}[h]
%%\begin{center}
%%\begin{psfrags}
%%\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
%%\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
%%\psfrag{1}{$a=bcb^{-1}$}
%%\psfrag{3}{$c=aba^{-1}$}
%%\includegraphics{trefoil_group.eps}
%%\end{psfrags}
%%\end{center}
%%\end{figure}
%
  \only<1>{
    \begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group1.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }
  \only<2>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group2.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }

  \only<3>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group3.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }

    \only<4>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group4.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }

    \only<5>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group5.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}  }
  \only<6>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group6.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }
  \only<7>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group7.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
\textcolor[rgb]{1.00,1.00,1.00}{Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$}
\end{center}
  }
\only<8>{
\begin{figure}
\begin{center}
\begin{psfrags}
\psfrag{a}{$a$}\psfrag{b}{$b$}\psfrag{c}{$c$}
\psfrag{T}{$T$}\psfrag{2}{$b=cac^{-1}$}
\psfrag{1}{$a=bcb^{-1}$}
\psfrag{3}{$c=aba^{-1}$}
\includegraphics{pi1/trefoil_group7.eps}
\end{psfrags}
\end{center}
\end{figure}
\begin{center}
Thus, $b = aba^{-1} a a b^{-1}a^{-1}$ or $bab = aba$
\end{center}
}
%
%
\end{frame}



\begin{frame}
 \frametitle{Computing the Knot group}
  \framesubtitle{The Trefoil Group}

The computations above show:

\vfill

\begin{itemize}

\item<2-> $\pi_1 (\R^3-T) = \langle a,b |~aba = bab\rangle$

\vfill

\item<3-> \textbf{Fact:} $\pi_1 (\R^3-T)$ is \textbf{not} Abelian

\vfill

\item<4-> \textbf{Fact:} $K$ is the unknot, $\pi_1(\R^3-K) = \Z$

\vfill

\item<5->
\begin{center}
Unknot $\neq$ Trefoil!
\end{center}


\end{itemize}

\end{frame}




%\subsection{Alexander Polynomials}
%
%\begin{frame}
%\frametitle{Alexander Polynomial}
% \framesubtitle{Separation Properties}
%
%
%\begin{lemma} A Diagram with $c$ crossings separates the plane in $c+2$ regions.
%\end{lemma}
%
%\vfill
%
%\uncover<2->{\emph{Proof}: The Euler characteristic of $S^2$ is 2.}
%
%\vfill
%
%\uncover<3->{\begin{project}
%Learn about Classification of Surfaces via the Euler Characteristic and/or understand the Gauss-Bonnet
%\[
%\int_M K~dA = 2\pi\chi(M)
%\]
%\end{project}}
%
%\vfill
%
%\uncover<4->{
%\begin{itemize}
%\item Topology ``detects'' Geometry.
%\end{itemize}}
%
%\end{frame}








\end{document} 