introduction.tex

\chapter{Introduction to Knot Theory}

\section{Definition of a knot}

We normally conceive of knots as open strings with `knotted' parts in between. Given a knotted string with two `open' ends, we can simply pull one end of the string to untie it, in the usual way we untie a knot, by inserting an open end into the knotted region is a strategic manner and pulling it on the other side\footnote{There exists another way of unknotting an open knot, often used in magic tricks. One creates another knot using the unknotted portion of the string on one side of the knotted portion such that this new knot `cancels' the original knot. In mathematical terms, the new knot is constructed such that the \textit{connected sum} of the original knot and the new knot gives an unknotted circle, or the \textit{unknot}. We shall not deal with connected sums of knots in this thesis.}. This way, any knot with two open ends can be untied. But if have a knot in a closed loop, we won't be able to untie it unless we cut it. We want our notion of a knot to be invariant under `pulling'. Thus, we model our mathematical definition on closed loops instead of open. Refer to~\cref{fig:openclosedknots}.

\begin{figure}
    \centering
    \subcaptionbox{A knot with open ends}{
        \begin{tikzpicture}[scale=0.2]
            \begin{knot}[clip width = 8, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                \strand[thick, red] (-6,0)
                to (6,0)
                %to [out=0, in=135] (13,-2)
                to [out=0, in=180] (18, -4)
                to [out=0, in=-90] (21,0)
                to [out=90, in=0] (18,4)
                to [out=180, in=0] (9,-4)
                to [out=180, in=-90] (6,0)
                to [out=90, in=180] (9,4)
                to [out=0, in=180] (21,0)
                to (33,0);
                \flipcrossings{1,2,4,5,6,7,8,9}
            \end{knot}
        \end{tikzpicture}}\par\bigskip
    \subcaptionbox{A knot with closed ends}{
        \begin{tikzpicture}[scale=0.2]
            \begin{knot}[clip width = 8, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                \strand[thick, red] (2,0) to (6,0)
                to [out=0, in=180] (18, -4)
                to [out=0, in=-90] (21,0)
                to [out=90, in=0] (18,4)
                to [out=180, in=0] (9,-4)
                to [out=180, in=-90] (6,0)
                to [out=90, in=180] (9,4)
                to [out=0, in=180] (21,0)
                to (25,0);
                \flipcrossings{4,5,7,8,9,11,12}
            \end{knot}
            \draw[thick, red, rounded corners] (2,0) -- (1,0) -- (0,-1) -- (0,-5) -- (1,-6) -- (26,-6) -- (27,-5) -- (27,-1) -- (26,0) -- (25,0);
%             \draw[thick, red, rounded corners=9] (1,-6) -- (0,-6) -- (0,0) -- (1,0);
        \end{tikzpicture}}
    \caption{Projections of a knot on a plane}
    \label{fig:openclosedknots}
\end{figure}

\begin{defn}[Knot]
    A knot \(K\) is the image a topological embedding of \(\sone\) in \(\rthree\).
\end{defn}

\begin{figure}
    \centering
    \begin{tikzpicture}[scale=0.2]
        \begin{knot}[clip width = 8, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
            \strand[thick, red] (1,0)
            to (6,0)
            to [out=0, in=180] (18, -4)
            to [out=0, in=-90] (21,0)
            to [out=90, in=0] (18,4)
            to [out=180, in=0] (9,-4)
            to [out=180, in=-90] (6,0)
            to [out=90, in=180] (9,4)
            to [out=0, in=180] (21,0)
            to (22.5,0);
            \flipcrossings{1,2,4,5,6,7,8,9}
        \end{knot}
            \begin{scope}[xshift=23cm, scale=0.6]
                \begin{knot}[clip width = 6, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                    \strand[thick, red] (-2,0)
                    to (6,0)
                    to [out=0, in=180] (18, -4)
                    to [out=0, in=-90] (21,0)
                    to [out=90, in=0] (18,4)
                    to [out=180, in=0] (9,-4)
                    to [out=180, in=-90] (6,0)
                    to [out=90, in=180] (9,4)
                    to [out=0, in=180] (21,0)
                    to (23,0);
                    \flipcrossings{1,2,4,5,6,7,8,9}
                \end{knot}
            \end{scope}
                \begin{scope}[xshift=37cm, scale=0.35]
                    \begin{knot}[clip radius =3pt, clip width = 3, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                        \strand[thick, red] (-2,0)
                        to (6,0)
                        to [out=0, in=180] (18, -4)
                        to [out=0, in=-90] (21,0)
                        to [out=90, in=0] (18,4)
                        to [out=180, in=0] (9,-4)
                        to [out=180, in=-90] (6,0)
                        to [out=90, in=180] (9,4)
                        to [out=0, in=180] (21,0)
                        to (23,0);
                        \flipcrossings{1,2,4,5,6,7,8,9}
                    \end{knot}
                \end{scope}
                    \begin{scope}[xshift=45cm, scale=0.2]
                        \begin{knot}[clip radius =3pt, clip width = 1.7, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                            \strand[thick, red] (-2,0)
                            to (6,0)
                            to [out=0, in=180] (18, -4)
                            to [out=0, in=-90] (21,0)
                            to [out=90, in=0] (18,4)
                            to [out=180, in=0] (9,-4)
                            to [out=180, in=-90] (6,0)
                            to [out=90, in=180] (9,4)
                            to [out=0, in=180] (21,0)
                            to (22,0);
                            \strand [thick, red, only when rendering/.style={densely dotted}] (22,0) to (46,0);
                            \flipcrossings{1,2,4,5,6,7,8,9}
                        \end{knot}
                    \end{scope}
        \draw[thick, red, rounded corners] (1,0) to (0,0) to (0,-6) to (55,-6) to (55,0) to (53,0);
        \node (p) at (53,1.5) {\(p\)};
        \filldraw[black] (53,0) circle (5pt);
    \end{tikzpicture}
    \caption{A wild knot. The knot is wild only at the point \(p\).}
    \label{fig:wildknot}
\end{figure}

Although one can talk about embeddings of higher dimensional `circles' in higher dimensional spaces, we restrict ourselves to knots in the three dimensional space. The above definition turns out out to be too general for our purposes. It takes into consideration certain pathological knots known as \textit{wild knots}. Although wild knots are an object of study, we shall not be dealing with them in this thesis due to their pathological nature. \Cref{fig:wildknot} shows an example of a wild knot. A section of the knot is scaled down by a factor and then joined on one side of the previous section. If one repeats this process infinitely, the resulting curve formed by appending sections eventually converges to a point, provided that the scaling down is fast enough. We can join the convergence (limit) point to a an end-section on the other side to get wild knot. This knot is continuous everywhere, including the limit point. The behaviour of the knot at the limit point is different from the other points. Three common ways exist to exclude such behaviour by demanding extra conditions.
\begin{enumerate}
    \item Differentiability.

    We can demand all knots to be differentiable at all points. We can verify visually that all the points, except the limit point of the wild knot are differentiable, or can be made (isotoped) differentiable. The derivative must necessarily change within a section. As the size of the section decreases, the derivative changes more rapidly. In a scaled down section, all the values which the derivative took in the previous section. As we approach the sections near the limit point, the derivative function must attain all the values it did before, but it must do so in a more rapid manner. At the limit point, the derivative shall cease to exist by the virtue of `changing too rapidly'. Demanding differentiability forcibly removes the offending limit point. The wildness is due to the limit point. Along with differentiability, we can demand \(\symup{C}^r\) smoothness or \(\symup{C}^\infty\) smoothness as well.

    But this condition comes with a problem as well, namely we cannot use polygons for describing knots.
    \item Piecewise linearity.

    We can demand all the edges of a knot to be piecewise linear with finitely many edges. Our knot shall be a polynomial (with finitely many edges) in that case. Infinitely (countably) many sections in a wild knot shall mean infinitely many edges (of decreasing length), which is not allowed. Thus, this condition excludes wild knots.
    \item Local flatness.
\end{enumerate}
In this thesis, we shall take the third route following Cromwell~\cite[chp.~1]{cromwell}. Local flatness is a topological condition unlike other other two. If we consider a local neighbourhood around each point of the knot, except the limit point, then we see visually see that we can always find a small enough local neighbourhood around each such point such that the strand is not `knotted' in that neighbourhood. At the limit point, no matter how small a neighbourhood we take, the strand shall always be `knotted'. We enforce this `local unknottedness' condition by demanding local flatness. Let \(p\) be a point in a knot \(K\), \(\symup{B}(\symup{O}, 1)\) be the unit ball centered at origin \(\symup{O}\) and \(d\) be a diameter of \(\symup{B}(\symup{O}, 1)\).
\begin{defn}[Local flatness]
    The point \(p\) is said to be locally flat if there exists a neighbourhood \(U \ni p\) such that the pair \((U, U \cap K)\) is homeomorphic to \((\symup{B}(\symup{O}, 1), d)\).

    A knot is said to be locally flat if each point in that knot is locally flat. A point that is not locally flat is called wild, and a knot is wild if any of its points are wild.
\end{defn}

Consider a spherical neighbourhood around a locally flat point. There exists a radius such that for all neighbourhoods less than this radius, the boundary of the neighbourhood intersects the strand in exactly two points. This is not possible at the limit point in wild knot figure.

\begin{defn}[Tame knots]
    A knot is said to be tame if all its points are locally flat.
\end{defn}

\section{Distinguishing knots}

Any two homeomorphisms of the circle are homeomorphic to the circle and to each other, since being homeomorphic is an equivalence relation. But this means that all knots are homeomorphic to each other. Clearly, homeomorphism is not the correct notion to distinguish knots. When we mean that two knots are distinct, we mean that if we create a physical model of those knots, we cannot `physically deform' one knot into another. \textit{Cutting a knot is not allowed.} One might think that homotopy or isotopy are what we need, but it turns out that the notion of \textit{ambient isotopy} is the correct one.

\begin{defn}[Homotopy]
    A homotopy of a space \(X \subset \rthree\) is a continuous map \(h \colon X \times [0, 1] \rightarrow \rthree\).

    The restriction of \(h\) to level \(t \in [0, 1]\) is \(h_t \colon X \times \{t\} \rightarrow \rthree\). \(h_0\) must be the identity map.
\end{defn}
Note that the continuity of \(h\) implies the continuity of \(h_t\) for all \(t \in [0,1]\). The converse is not true though. Homotopy allows a curve to pass through itself. All knots are thus homotopic to the unknot, also referred to as the trivial knot. If we do not allow a curve to pass through itself, i.e.\@ if we demand injectivity for each \(h_t\), then we get what is known as an isotopy. But isotopy is not useful for distinguishing knots as well, due to bachelors' unknotting. All tame knots, or more generally, all knots with a tame arc turn out to be isotopic to the trivial knot. It is an open problem if all knots are isotopic to the trivial knot~\cite{ancel, shijie}. In addition, it is not known as well if a knot known as the Bing sling, which is wild \textit{at all points} is isotopic to the trivial knot~\cite{ancel, shijie}.
\begin{prop}[Bachelors' unknotting]
    Every knot with a tame arc is isotopic to the trivial knot.
\end{prop}
\begin{proof}
    \begin{figure}
        \centering
        \subcaptionbox{\(i_0 (K)\)}{
            \begin{tikzpicture}[scale=0.2]
                \begin{knot}[clip width = 10, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                    \strand[thick, red] (1,0) to (6,0)
                    to [out=0, in=180] (18, -4)
                    to [out=0, in=-90] (21,0)
                    to [out=90, in=0] (18,4)
                    to [out=180, in=0] (9,-4)
                    to [out=180, in=-90] (6,0)
                    to [out=90, in=180] (9,4)
                    to [out=0, in=180] (21,0)
                    to (26,0);
                    \flipcrossings{4,5,7,8,9,11,12}
                \end{knot}
                \draw[thick, red, rounded corners] (26,0) -- (27,0) -- (27,-6) -- (0,-6) -- (0,0) -- (1,0);
                \filldraw[black] (13.5,-6) circle (5pt) node[anchor=north, yshift=-0.3cm]{\(p\)};
                \draw[densely dashed, line width=0.6pt, rounded corners, closed] (3,-1) to (3,1) to (-1,1) to (-1,-7) to (28,-7) to (28,1) to (24,1) to (24,-1) to (26,-1) to (26,-5) to (1,-5) to (1,-1) -- cycle;
                \node at (26.4,2.5) {\(U_p\)};
            \end{tikzpicture}}
        \subcaptionbox{\(i_{0.3} (K)\)}{
            \begin{tikzpicture}[scale=0.2]
                \begin{scope}[xshift=4cm, scale=0.7]
                    \begin{knot}[clip width = 7, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                        \strand[thick, red] (0,0) to (6,0)
                        to [out=0, in=180] (18, -4)
                        to [out=0, in=-90] (21,0)
                        to [out=90, in=0] (18,4)
                        to [out=180, in=0] (9,-4)
                        to [out=180, in=-90] (6,0)
                        to [out=90, in=180] (9,4)
                        to [out=0, in=180] (21,0)
                        to (27,0);
                        \flipcrossings{4,5,7,8,9,11,12}
                    \end{knot}
                \end{scope}
                \draw[thick, red, rounded corners, rounded corners] (22,0) to (27,0) to (27,-6) to (0,-6) -- (0,0) -- (5,0);
                \filldraw[black] (13.5,-6) circle (5pt) node[anchor=north, yshift=-0.3cm]{\(p\)};
                \draw[densely dashed, line width=0.6pt, rounded corners] (3,-1) to (3,1) to (-1,1) to (-1,-7) to (28,-7) to (28,1) to (24,1) to (24,-1) to (26,-1) to (26,-5) to (1,-5) to (1,-1) -- cycle;
                \node at (26.4,2.5) {\(U_p\)};
            \end{tikzpicture}}
        \subcaptionbox{\(i_{0.6} (K)\)}{
            \begin{tikzpicture}[scale=0.2]
                \begin{scope}[xshift=8cm, scale=0.4]
                    \begin{knot}[clip radius = 2pt, clip width = 4, consider self intersections = true, ignore endpoint intersections = false, use Hobby shortcut]
                        \strand[thick, red] (0,0) to (6,0)
                        to [out=0, in=180] (18, -4)
                        to [out=0, in=-90] (21,0)
                        to [out=90, in=0] (18,4)
                        to [out=180, in=0] (9,-4)
                        to [out=180, in=-90] (6,0)
                        to [out=90, in=180] (9,4)
                        to [out=0, in=180] (21,0)
                        to (30,0);
                        \flipcrossings{4,5,7,8,9,11,12}
                    \end{knot}
                \end{scope}
                \draw[thick, red, rounded corners] (19,0) to (27,0) to (27,-6) to (0,-6) to (0,-6) -- (0,0) -- (8.1,0);
                \filldraw[black] (13.5,-6) circle (5pt) node[anchor=north, yshift=-0.3cm]{\(p\)};
                \draw[densely dashed, line width=0.6pt, rounded corners] (3,-1) to (3,1) to (-1,1) to (-1,-7) to (28,-7) to (28,1) to (24,1) to (24,-1) to (26,-1) to (26,-5) to (1,-5) to (1,-1)  -- cycle;
                \node at (26.4,2.5) {\(U_p\)};
            \end{tikzpicture}}
        \subcaptionbox{\(i_1 (K)\)}{
            \begin{tikzpicture}[scale=0.2]
                \draw[thick, red, rounded corners] (22,0) to (27,0) to (27,-6) to (0,-6) -- (0,0) -- (22,0);
                \filldraw[black] (13.5,-6) circle (5pt) node[anchor=north, yshift=-0.3cm]{\(p\)};
                \draw[densely dashed, line width=0.6pt, rounded corners] (3,-1) to (3,1) to (-1,1) to (-1,-7) to (28,-7) to (28,1) to (24,1) to (24,-1) to (26,-1) to (26,-5) to (1,-5) to (1,-1) -- cycle;
                \node at (26.4,2.5) {\(U_p\)};
            \end{tikzpicture}}
        \caption{Bachelors' unknotting demonstrating the isotopy equivalence of a knot with tame arc to the trivial knot. Here we have chosen the point \(r\) (not shown in the figure) to be the midpoint of the knotted region. \(f(a)\) and \(f(b)\) are the intersections of the knot \(K\), represented by the thick red line, with \(U_p\), shown using densely dashed lines. Note that \(f(a)\), \(r\) and \(f(b)\) are collinear in this case. The entirety of the knot is tame in this case, but we only require tameness in the chosen region of \(U_p \cap K\).}
        \label{fig:bachelor}
    \end{figure}

    Refer to \cref{fig:bachelor}. Let \(K \subset \rthree\) be a knot with a tame arc.

    Let \(p \in K\) in a tame arc of the knot. Since the knot is locally flat on the arc, by the definition of tameness, we take a ball \(U_p \subset \rthree\) of radius \(\varepsilon\) around the point \(p\) such that the pair \(U_p, U_p \cap K\) is homeomorphic to \((\symup{B}, \symup{d})\), where \(\symup{B}\) is the unit ball in \(\rthree\) centered at the origin and \(\symup{d}\) is the diameter of \(\symup{B}\) along the \(x\)-axis. We choose a parametrization \(f \colon [0, 2\pii) \rightarrow K\) of the knot such that \(f([a, b]) = K - (U_p \cap K) \), where \([a, b] \subset [0, 2\pii)\).

    Let \(r \in \rthree\) be a point outside \(U_p\). Now consider the function \(i_t \colon K \rightarrow \rthree\), defined for each \(t \in [0, 1]\) as follows.
    Let \[\phi_t(a) \coloneq t\left(\frac{a+b}{2}\right) + (1-t)a\] be a family of functions for all \(t \in [0,1]\).
    \begin{enumerate}
        \item If \(f(x) \in U_p\), then \[i_t(f(x)) = f(x).\]
        \item If \(\displaystyle x \in [a, \phi_t(a))\), then \[ i_t(f(x)) = f(a) + \frac{\norm{f(a) - f(\phi_t(a))}}{\norm{a - \phi_t(a)}}(x - a)(r - f(a)).\]
        \item If \(x \in [\phi_t(a), \phi_t(b)]\), then \[i_t(f(x)) = tr + (1-t)f(x).\]
        \item If \(\displaystyle x \in (\phi_t(b), b]\), then \[ i_t(f(x)) = f(b) + \frac{\norm{f(b) - f(\phi_t(b))}}{\norm{b - \phi_t(b)}}(b - x)(r - f(b)).\]
    \end{enumerate}
    Let \(i \colon [0, 1] \times K \rightarrow \rthree\) be a function defined by \(i(t, f(x)) \coloneq i_t (f(x))\).

    \(i\) is defined such that the part inside \(U_p\) is kept unchanged for all \(t\). For \(t =0\), \(i\) does not deform the knot at all. For \(t \in (0, 1)\), the knotted part (in \(\rthree\)) shrinks and the interval in the domain \([a,b]\) which maps to the knotted part also shrinks to \([\phi_t(a), \phi_t(b)]\). This shrinkage of the domain happens linearly with \(t\). Refer to \cref{fig:graphbachelor}. All points of the knotted part trace a straight line as they travel under isotopy from their original position to \(r\). Eventually, the knotted part ceases to exist at \(t = 1\) and a single point of the domain, namely, \((a + b)/2\) maps to \(r\).

    \begin{figure}
        \centering
        \begin{tikzpicture}[scale=1.4]
            \draw[thick, blue, <->] (0,4.5) node[anchor=south]{\(x\)} -- (0,0) -- (5,0);
            \draw[line width=0.6pt, black] (-2.5pt,4) node[anchor=east, black]{\(2\pii\)} -- (2.5pt,4);
            \node at (0,-0.2) {\(0\)};
            \node at (4,-0.2) {\(1\)};

            \draw[black, dotted, line width=0.6pt] (0,2) node[anchor=east, black]{\((a+b)/2\)} -- (4,2);
            \draw[black, dotted, line width=0.6pt] (2,0) -- (2,4);
            \draw[black, densely dashed, line width=0.7pt] (2,1.5) -- (0,1.5) node[anchor=east, black]{\(\phi_t(a)\)};
            \draw[black, densely dashed, line width=0.7pt] (2,2.5) -- (0,2.5) node[anchor=east, black]{\(\phi_t(b)\)};
%             \node at (-0.36,1.5) {\(\phi_t(a)\)};
%             \node at (-0.36,2.5) {\(\phi_t(b)\)};
%             \node at (-0.6,1.98) {\((a+b)/2\)};
            \node at (2,-0.2) {\(t\)};
            \draw[thick, red] (0,1) node[anchor=east, black]{\(a\)} -- (4,2) -- (0,3) node[anchor=east, black]{\(b\)};
        \end{tikzpicture}
        \caption{Illustration of shrinkage of domain with respect to \(t\). For all \(t \in [0,1]\), the interval \([\phi_t(a), \phi_t(b)] \in [0,2\pii)\) maps to the shrunk `knotted' part. The images of \([a,\phi_t(a)]\) and \([\phi_t(b),b]\) map to the straight lines which connect \(f(a)\) to \(i_t(f(\phi_t(a)))\) and \(i_t(f(\phi_t(b)))\) to \(f(b)\) respectively.}
        \label{fig:graphbachelor}
    \end{figure}

    In the end, we get a figure consisting of two straight lines meeting at \(r\), and \(U_p \cap K\), the original part of the knot inside \(U_p\). The other endpoints of these lines are \(f(a)\) and \(f(b)\). \(U_p \cap K\) is isotopic to the line joining \(f(a)\) and \(f(b)\). Thus, we get a triangle with points \(r\), \(f(a)\) and \(f(b)\) and we any that any triangle in \(\rthree\) is isotopic to \(\sone\).

    We now prove that \(i\) is continuous. We know that both \(i_t\) and \(i_x\) are continuous and injective for all \(t \in [0, 1]\) and \(x \in [0, 2\pii)\) respectively, where \(i_x\) is defined to be the restriction of \(i\) for a particular \(x\). We also see that \(i_t\) is linear in \(x\). A function continuous in one argument and linear in the other is continuous in the product topology. We thus have an isotopy which sends a knot with a tame arc to \(\sone\).
\end{proof}

\begin{remark}
    It should be noted that the isotopy that we have constructed does not shrink \(K - (U_p \cap K)\) uniformly. Parts of the strand closer to the point \(r\) are shrunk more than the parts further away. Also, not all parts move at a uniform rate towards \(r\). Parts closer to \(r\) move slower than the parts further.%\footnote{The knotted parts are scaled uniformly though in \cref{fig:bachelor} due to the ease of drawing such a figure.}.
\end{remark}


In the above considered isotopy, we isotoped the set \(X = K\). Instead, we take \(X\) to be the entire space \(\rthree\), or a bounded set which completely covers the knot, then we get the notion of ambient isotopy. This modification ensures that the surrounding space is isotoped as well as we isotope the knot. The knot is a curve which has no volume. If we try bachelors' unknotting on the surrounding space as well, we observe that the surrounding space, which has a finite, non-zero volume cannot shrink to a set of zero volume under isotopy. This finally leads us to the equivalence relation induced by ambient isotopy.

\begin{remark}
    Unless mentioned otherwise, we shall always consider our knots to be tame from now on.
\end{remark}

\begin{defn}[Knot equivalence]
    Two knots \(K_1\) and \(K_2\) are said to be ambient isotopic if there exists an isotopy \(I \colon \rthree \times [0,1] \rightarrow \rthree\) such that \(I(K_1,0) = I_0(K_1) = K_1\) and \(I(K_1,1) = I_1(K_1) = K_2\).
\end{defn}
Knot equivalence is an equivalence relation as it satisfies the properties of reflexivity, symmetry and transitivity.

\begin{remark}
    Each equivalence class of knots is called a \textit{knot type}. We would often forget the distinction between a knot and its knot type. The intended meaning can be inferred from the context.
\end{remark}

\begin{remark}
    Note that we distinguish between \textit{ambient isotopy} and \textit{isotopy}. Many treatments of knot theory use the word isotopy for ambient isotopy as ambient isotopy is the useful construct in knot theory. Ambient isotopy is an isotopy of the whole space containing the knot, not just the knot.
\end{remark}

\begin{remark}
    All three approaches for excluding pathological behaviour, including \(\symup{C}^r\) for all \(r \geq 1\), generate the same knot isotopy classes. The knot isotopy classes corresponding to \(\symup{C}^1\) curves, piecewise-linear curves with finitely many components, smooth \(\symup{C}^\infty\) curves and locally flat everywhere curves are equal~\cite[\S~1.11]{cromwell}.
\end{remark}

\begin{exmp}
    The knots depicted in \cref{fig:variousknots} are non-trivial and not ambient isotopic to each other. We shall prove this fact by the use of Kauffman's bracket polynomial in later chapters.
    \begin{figure}
        \centering
        \subcaptionbox{A trefoil}{
            \begin{tikzpicture}[
                use Hobby shortcut,
                every trefoil component/.style={thick, draw},
                trefoil component 1/.style={red},
                trefoil component 2/.style={red},
                trefoil component 3/.style={red},
                ]
                \path[spath/save=trefoil] ([closed]90:2) foreach \k in {1,...,3} {
                    .. (-30+\k*240:.5) .. (90+\k*240:2) } (90:2);
                \tikzset{spath/knot={trefoil}{8pt}{1,3,5}}
            \end{tikzpicture}}
        \subcaptionbox{The figure-eight knot}{
            \begin{tikzpicture}[thick, red, every path/.style={red,thick}, every
                node/.style={transform shape, knot crossing, inner sep=1.5pt}]
                \node[rotate=45] (tl) at (-1,1) {};
                \node[rotate=-45] (tr) at (1,1) {};
                \node (m) at (0,-1) {};
                \node (b) at (0,-2) {};
                \draw[thick, red] (b) .. controls (b.4 north west) and (m.4 south west) ..
                (m.center);
                \draw[thick, red] (b.center) .. controls (b.4 north east) and (m.4 south east)
                .. (m);
                \draw[thick, red] (m) .. controls (m.8 north west) and (tl.3 south west) ..
                (tl.center);
                \draw[thick, red] (m.center) .. controls (m.8 north east) and (tr.3 south
                east) .. (tr);
                \draw[thick, red] (tl.center) .. controls (tl.16 north east) and (tr.16 north
                west) .. (tr);
                \draw[thick, red] (b) .. controls (b.16 south east) and (tr.16 north east) ..
                (tr.center);
                \draw[thick, red] (b.center) .. controls (b.16 south west) and (tl.16 north
                west) .. (tl);
                \draw[thick, red] (tl) -- (tr.center);
            \end{tikzpicture}}
        \subcaptionbox{The cinquefoil knot}{
            \begin{tikzpicture}[rotate=18]
                \begin{knot}[thick, red, clip width=7,
                    consider self intersections=true,
                    %  draft mode=crossings,
                    flip crossing/.list={2,4},
                    only when rendering/.style={
                        %    show curve controls
                    }
                    ]
                    \strand[thick, red] (2,0) .. controls +(0,1.0) and +(54:1.0) .. (144:2) .. controls +(54:-1.0) and +(18:-1.0) .. (-72:2) .. controls +(18:1.0) and +(162:-1.0) .. (72:2) .. controls +(162:1.0) and +(126:1.0) .. (-144:2) .. controls +(126:-1.0) and +(0,-1.0) .. (2,0);
                \end{knot}
            \end{tikzpicture}}
        \quad\quad\subcaptionbox{The three-twist knot (\(5_2\))}{
            \begin{tikzpicture}[use Hobby shortcut]
                \begin{knot}[
                    clip width=7,
                    consider self intersections=true,
                    %  draft mode=crossings,
                    ignore endpoint intersections=false,
                    flip crossing/.list={6,4,2}
                    ]
                    \strand[thick, red] ([closed]2,2) .. (1.8,0) .. (-2.3,-1) .. (.5,1) .. (-2,2) .. (-1.8,0) .. (2.3,-1) .. (-.5,1) .. (2,2);
                \end{knot}
            \end{tikzpicture}}
        \caption{Various distinct knots.}
        \label{fig:variousknots}
    \end{figure}
\end{exmp}

\begin{remark}
    In this thesis, we shall look at knot theory in \(\rthree\). One can compactify \(\rthree\) to \(\sthree\) and do knot theory in \(\sthree\), as many treatments do. This does not result in a different knot theory.
\end{remark}

\section{Links}

So far, we have looked at embeddings of a single circle. We can embed more than one circles to get links.

\begin{defn}
    A collection of disjoint topological embeddings of \(\sone\) in a three dimensional space is called a link.
\end{defn}

Each knot (an embedding of a circle) belonging to a link is called a \textit{component} of the link. The number of components of a link is called the \textit{multiplicity} of the link. We shall denote the multiplicity of a link \(L\) by \(\mu(L)\). A knot is thus a link with one component.

A link such that when projected onto a plane gives us \(n\) disjoint circles is referred to as a trivial link with \(\mu(L) = n\) components. Refer to \cref{fig:triviallink} for a link with \(n = 4\). A link is trivial iff all its components are trivial knots and they are `unlinked'. Equivalently, a link is trivial if its components bound disjoint discs.

\begin{figure}
    \centering
    \begin{tikzpicture}
        \draw[thick, red] (0,0) circle [radius=1];
        \draw[thick, red] (2.5,0) circle [radius=1];
        \draw[thick, red] (5,0) circle [radius=1];
        \draw[thick, red] (7.5,0) circle [radius=1];
    \end{tikzpicture}
    \caption{A trivial link with \(4\) components}
    \label{fig:triviallink}
\end{figure}

Since a link is a union of knots, we might try to describe a link \(L\) as \(L = K_1 \cup \cdots \cup K_n\) such that images of \(K_i\)'s are disjoint. But this does not completely describe a link as this only lists the component parts, and does not indicate how they are put together in space.

We can define two links to be equivalent if they are ambient isotopic, i.e.\@ there exists an ambient isotopy which takes one link to another. This equivalence relation is \textit{weak} in the sense that one has the freedom to match different components of the two links. We have not labelled the links.

\begin{figure}
    \centering
    \begin{tikzpicture}
        \begin{knot}[clip width=7]
            \strand[thick, red] (0,0) circle [radius=1];
            \strand[thick, blue] (1,0) circle [radius=1];
            \flipcrossings{1}
        \end{knot}
    \end{tikzpicture}
    \caption{A Hopf link}
    \label{fig:hopflink}
\end{figure}

\section{Link diagrams}

So far, we have depicted knots as curves in a plane (of paper) with the implicit understanding that if a strand goes under another strand, then we break the strand which goes underneath. We defined a knot as an object in three dimensions, but we can project the knot onto a plane to represent it, as we have done so far. We do not lose any topological information if we choose a `nice enough' projection. What we have been doing implicitly can be formalized; we shall see how. We shall introduce some preliminary topological notions which shall be required.

\subsection{Topological manifolds}

\begin{defn}[Topological manifold]
    A topological \(n\)-manifold is a Hausdorff and second-countable topological space such that each point has a neighbourhood homeomorphic to \(\R^n\) or \(\R^n_{\geq 0}\).
\end{defn}

\begin{remark}
    Different sources define a topological manifold in slightly differing ways. Some require just paracompactness instead of second-countability. The above definition includes manifolds with boundary as well. Boundary points shall be the points which map to the boundary of \(\R^n_{\geq 0}\).
\end{remark}

\begin{defn}
    A curve is a compact \(1\)-manifold and a surface is a compact \(2\)-manifold.
\end{defn}

There are only four distinct connected \(1\)-manifolds, namely the real line, the real half-line, the compact interval and the circle u p to homeomorphism.

\subsection{Transversality}

We shall need to consider the ways surfaces and curves intersect each other. We use the principle of general position to classify these intersections. We shall follow the treatment of Cromwell in this section~\cite[chp.~2, \S~2.10]{cromwell}.

A structure or property is said to be \textit{stable} if it maintains its essential characteristics when it is perturbed a little. In our context, this means that the original set of curves and surfaces must be homeomorphic to the perturbed set of curves and surfaces. We desire our intersections to be stable. A collection of curves in a plane is said to be \textit{stable} if the collection maintains its essential characteristics when perturbed a little. For example, consider the intersection of circle and a line as shown in \cref{fig:intersectionoflineandcircle}.

\begin{figure}
    \centering
    \subcaptionbox{\label{subfig:intersectiona}}{\CWL}\quad\quad
    \subcaptionbox{\label{subfig:intersectionb}}{\CWLO}\quad\quad
    \subcaptionbox{\label{subfig:intersectionc}}{\CWLI}
    \caption{Intersection of a line and a circle}\label{fig:intersectionoflineandcircle}
\end{figure}

The intersection as shown in \cref{subfig:intersectiona} is not stable as any translation or rotation of the straight line shall lead to a figure homeomorphic to \cref{subfig:intersectionb} or \cref{subfig:intersectionc}. The figure does not represent a \(1\)-manifold as no neighbourhood around the intersection point is homeomorphic to \(\R^n\). \Cref{subfig:intersectionb} is a union of two connected \(1\)-manifold components. \Cref{subfig:intersectionc} is again not a \(1\)-manifold. Since the number of intersection points is different in each case, all three figures are topologically distinct (not homeomorphic). The intersections in \cref{subfig:intersectionb} and \cref{subfig:intersectionc} are stable though.

\begin{defn}
    An intersection is said to be \textit{transverse} if a neighbourhood of it is homeomorphic to one of the following four cases.
    \begin{enumerate}
        \item The union of \(x\)-axis and \(y\)-axis in \(\R^2\).
        \item The union of \(z\)-axis and \(xy\)-plane in \(\R^3\).
        \item The union of \(xz\)-plane and \(yz\)-plane in \(\R^3\).
        \item The union of \(xy\)-plane, \(xz\)-plane and \(yz\)-plane in \(\R^3\).
    \end{enumerate}
\end{defn}

\begin{thm}
    The above four cases the only stable arrangements of curves and surfaces in two and three dimensions.
\end{thm}

The above theorem states that if we have a stable set of curves and surfaces in two and three dimensions, then the set is homeomorphic to one the above four cases. A set of objects is said to be in \textit{general position} if all their intersections are transverse.

\begin{thm}
    Every finite set of curves and surfaces embedded in \(\R^3\) is arbitrarily close to an ambient isotopic set in general position.
\end{thm}

Thus, we can always eliminate points of tangency such as saddle points, maxima and minima. A detailed treatment about transversality of manifolds can be found in books of differential topology, such as the book of Guillemin and Pollack~\cite{pollack}.

\subsection{Projections and diagrams}

Let \(L\) be a link and let \(\pi \colon \rthree \rightarrow \rtwo\) be a projection map. A point \(p \in \pi(L)\) is called \textit{regular} if \(\pi^{-1}(p)\) is a single point, and \textit{singular} otherwise. If \(\abs{\pi^{-1}(p)} = 2\), then \(p\) is called a \textit{double} point.

We wish to get `nice enough' projections. There can be infinitely many singular points that occur in a projection. We demand that a projection contains only a finite number of singular points. Singular points shall be isolated in such a case. This shall exclude cases such as two line segments of link projecting onto the same line segment on the plane. The singular points can have multiple pre-images as well. We thus demand that \(\abs{\pi^{-1} (x)} \leq 2\). We also demand our projections to be stable so that if we change the projection direction by a small amount, the projection is essentially unchanged: no singular points are created or destroyed.

\begin{defn}
    If \(\pi(L)\) has a finite number of singular points and they are all transverse double points, then the projection is said to be a \textit{regular} projection.
\end{defn}

\begin{thm}
    Every tame link admits a regular projection.
\end{thm}

A link projection with the added information about the relative heights at crossings is called a link diagram. We adopt the convention that if a strand passes underneath another strand, then we break the underneath strand at the crossing in the diagram. A regular projection has only finitely many crossings.

\begin{thm}
    Every tame link admits a diagram.
\end{thm}

Proofs of the above two theorems can be found in Cromwell~\cite[chp.~3]{cromwell}. From now on, unless mentioned otherwise, we shall always assume that our link diagrams correspond to regular projections.

\begin{figure}
    \centering
    \subcaptionbox{A knot projection of the cinquefoil knot}[10cm]{
        \begin{tikzpicture}[rotate=18]
            \begin{knot}[thick, red, clip width=7,
                consider self intersections=false,
                %  draft mode=crossings,
                flip crossing/.list={2,4},
                only when rendering/.style={
                    %    show curve controls
                }
                ]
                \strand[thick, red] (2,0) .. controls +(0,1.0) and +(54:1.0) .. (144:2) .. controls +(54:-1.0) and +(18:-1.0) .. (-72:2) .. controls +(18:1.0) and +(162:-1.0) .. (72:2) .. controls +(162:1.0) and +(126:1.0) .. (-144:2) .. controls +(126:-1.0) and +(0,-1.0) .. (2,0);
            \end{knot}
        \end{tikzpicture}}\par\bigskip
    \subcaptionbox{A knot diagram of the cinquefoil knot}[10cm]{
        \begin{tikzpicture}[rotate=18]
            \begin{knot}[thick, red, clip width=7,
                consider self intersections=true,
                %  draft mode=crossings,
                flip crossing/.list={2,4},
                only when rendering/.style={
                    %    show curve controls
                }
                ]
                \strand[thick, red] (2,0) .. controls +(0,1.0) and +(54:1.0) .. (144:2) .. controls +(54:-1.0) and +(18:-1.0) .. (-72:2) .. controls +(18:1.0) and +(162:-1.0) .. (72:2) .. controls +(162:1.0) and +(126:1.0) .. (-144:2) .. controls +(126:-1.0) and +(0,-1.0) .. (2,0);
            \end{knot}
        \end{tikzpicture}}
\end{figure}

\section{Reidemeister theorem}

We can now encode all the topological information about a link in a two dimensional diagrammatic representation. If two links are ambient isotopic, then a natural question arises whether there exists relation between their diagrams. The famous and important theorem of Reidemeister answers this question in affirmative. Given a link diagram, we define three moves on local sections of a diagram, called as Reidemeister moves (\cref{fig:reidemeistermoves}). These moves are local, i.e.\@ we always perform these moves, or the ambient isotopies corresponding to these moves in a small enough neighbourhood such that everything is constant outside this neighbourhood. Such a neighbourhood exists as the knots are tame and the projections regular. moves preserve the link type.

\begin{figure}
    \centering
    \subcaptionbox{A type I move}{
        \begin{tikzpicture}
            \begin{knot}[scale=2]
                \strand[thick, red] (0,0) .. controls (0.3,0.5) .. (0,1);
            \end{knot}
            \begin{knot}[scale=1, yshift=1cm, xshift=2.5cm, clip width=8, ignore endpoint intersections=false, consider self intersections=true]
                \strand[thick, red] (-1,1) to[out=-90,in=-90] (0.5,0) to [out=90,in=90] (-1,-1);
            \end{knot}
        \end{tikzpicture}}\par\bigskip
    \subcaptionbox{A type II move}{
        \begin{tikzpicture}
            \begin{knot}[scale=2]
                \strand[thick, red] (0,0) .. controls (0.3,0.5) .. (0,1);
                \strand[thick, red, xshift=0.7cm, xscale=-1] (0,0) .. controls (0.3,0.5) .. (0,1);
            \end{knot}
            \begin{knot}[scale=2, xshift=1.4cm, clip width=8]
                \strand[thick, red] (0,0) .. controls (0.7,0.5) .. (0,1);
                \strand[thick, red, xshift=0.7cm, xscale=-1] (0,0) .. controls (0.7,0.5) .. (0,1);
            \end{knot}
        \end{tikzpicture}}\par\bigskip
    \subcaptionbox{A type III move}{
        \begin{tikzpicture}[scale=0.8]
            \begin{knot}[clip width=7]
                \strand[thick, red] (-1,1) to (2,-2);
                \strand[thick, red] (1,1) to (-2,-2);
                \strand[ultra thick, blue] (-2,-1) to (2,-1);
            \end{knot}
            \begin{knot}[yshift=-1cm, yscale=-1, xshift=6cm, clip width=7]
                \strand[thick, red] (-1,1) to (2,-2);
                \strand[thick, red] (1,1) to (-2,-2);
                \strand[ultra thick, blue] (-2,-1) to (2,-1);
            \end{knot}
        \end{tikzpicture}}
    \caption{The Reidemeister moves.}
    \label{fig:reidemeistermoves}
\end{figure}

We now state the Reidemeister theorem. It was proved by Kurt Reidemeister in 1927~\cite{reidemeister}, and by James Alexander and Garland Briggs in 1926~\cite{alexanderbriggs}.

\begin{thm}[Reidemeister]
    Two diagrams of a link are related by a finite sequence of Reidemeister moves.
\end{thm}

A proof of the above theorem can be found in the book of Murasugi on knots and links~\cite[chp.~4]{murasugi}.

\subsection{Crossing number}

Let \(c(D)\) denote the number of crossings in a diagram \(D\) of a link \(L\). The crossing number \(c(L)\) is is the minimum number of crossings in any diagram of the link.

\section{Orientation}

We can assign a direction or orientation to links. Take three points \(a\), \(b\) and \(c\) on a circle. We can define an orientation using a path such that one encounters the point \(a\) at the start, \(b\) in the middle and \(c\) at the end. If we transpose the points once, then we see that the orientation reverses. Even permutations of the set \(\{a,b,c\}\) can be identified with one orientation and the odd permutations with the other. \Cref{fig:oriented} shows a trefoil knot and a Hopf link with orientations.

\begin{figure}
    \centering
    \subcaptionbox{The oriented cinquefoil knot}{
        \begin{tikzpicture}[rotate=18]
            \begin{knot}[thick, red, clip width=7,
                consider self intersections=true,
                %  draft mode=crossings,
                flip crossing/.list={2,4},
                only when rendering/.style={
                    %    show curve controls
                }
                ]
                \strand[thick, red, draw=red,
                only when rendering/.style={
                    postaction=decorate,
                },
                decoration={
                    markings,
                    mark=at position 1 with {\arrowreversed{To}}
                }] (2,0) .. controls +(0,1.0) and +(54:1.0) .. (144:2) .. controls +(54:-1.0) and +(18:-1.0) .. (-72:2) .. controls +(18:1.0) and +(162:-1.0) .. (72:2) .. controls +(162:1.0) and +(126:1.0) .. (-144:2) .. controls +(126:-1.0) and +(0,-1.0) .. (2,0);
            \end{knot}
        \end{tikzpicture}\label{fig:orientedcinquefoilknot}}\quad\quad
    \subcaptionbox{An oriented Hopf link}[4cm]{
        \begin{tikzpicture}
            \begin{knot}[clip width=7]
                \strand[thick, red, draw=red,
                only when rendering/.style={
                    postaction=decorate,
                },
                decoration={
                    markings,
                    mark=at position 1 with {\arrowreversed{To}}
                }] (0,0) circle [radius=1];
                \strand[thick, blue, draw=blue,
                only when rendering/.style={
                    postaction=decorate,
                },
                decoration={
                    markings,
                    mark=at position 1 with {\arrowreversed{To}}
                }] (1,0) circle [radius=1];
                \flipcrossings{1}
            \end{knot}
        \end{tikzpicture}\label{fig:orientedhopflink}}
    \caption{Oriented links}
    \label{fig:oriented}
\end{figure}

It can be seen that with orientations added, each crossing can be categorised in to two types, positive and negative. Refer to \cref{fig:crossings}. We can distinguish between the types of these crossings in the following manner. Hold your right hand thumb along the oriented direction of the upper strand. If the fingers curl along the oriented direction, then we call it a positive crossing and assign it the value \(+1\). If the fingers curl in the opposite direction, then we call it a negative crossing at assign it \(-1\).

\begin{figure}
    \centering
    \subcaptionbox{Positive crossing \(+1\)}[4cm]{
        \begin{tikzpicture}[scale=1.2]
            \begin{knot}[clip width = 6]
                \strand[->, thick, red] (1,1) to (0,2);
                \strand[->, thick, red] (0,1) to (1,2);
                \flipcrossings{1}
            \end{knot}
        \end{tikzpicture}}\quad
    \subcaptionbox{Negative crossing \(-1\)}[4cm]{
        \begin{tikzpicture}[scale=1.2]
            \begin{knot}[clip width = 6]
                \strand[->, thick, red] (1,1) to (0,2);
                \strand[->, thick, red] (0,1) to (1,2);
            \end{knot}
        \end{tikzpicture}}
    \caption{Assignment of crossing values}
    \label{fig:crossings}
\end{figure}

\subsection{Writhe}

We define \textit{writhe} \(\w\) of a diagram \(D\) as the sum of all \(\epsilon(c)\) for all \(c \in D\). \[\w(D) = \sum_{c\in D} \epsilon(c).\] Note that writhe is a property of a diagram of a link, not the link itself. A link can have diagrams corresponding to all the possible writhes (the integers). We see this fact by applying the Reidemeister type I move repeatedly to generate a diagram with the desired writhe. A type I move can increase or decrease the writhe of a diagram depending upon the direction of twist we apply.

For example, the oriented cinquefoil knot in \cref{fig:oriented} has a writhe of \(-5\) as all the crossings are negative. Changing the orientation to another direction does not change the writhe of a knot as under an orientation change, the crossings preserve their signs. However, this is not true for links with more than one components, i.e.\@ links which are not knots. The writhe of the oriented Hopf link as shown in \cref{fig:oriented} is \(-2\), but changing the orientations of the individual components gives us different writhes.

It was conjectured for many years that minimal diagrams of a link, i.e.\@ diagrams with the minimal crossing number, had the same writhe. It was shown by Kenneth Perko in 1974 that two diagrams in the Rolfsen catalogue represented the same knot~\cite{perko}. They were thought to represent different knots as both were minimal crossing diagrams with different writhes. These knot diagrams go by the famous name of `Perko pair'.

\subsection{Linking number}

Let \(D\) be an oriented diagram of a \(2\)-component link \(K_1 \cup K_2\), and let \(D_i\) denote the component of \(D\) corresponding to \(K_i\). The crossings of \(D\) are of three types: \(D_1\) with itself, \(D_2\) with itself, and \(D_1\) with \(D_2\). We shall denote the last group by \(D_1 \cap D_2\). We define the \textit{linking number} of \(D_1\) with \(D_2\) as \[\operatorname{lk}(D_1, D_2) = \frac{1}{2} \sum_{c \in D_1 \cap D_2} \epsilon(c).\] As the name suggests, this quantity captures how many times one component `winds around' or `goes around' an other component. The linking number for the Hopf link in \cref{fig:oriented} is \(-1\). If we change the orientation of any of the components, keeping the other constant, we shall get a linking number of \(+1\). Changing the orientation of a component and keeping the others constant shall change only the sign of the linking number, and not the absolute value.

Linking number is an invariant for oriented two component links as it does not change under the Reidemeister moves. This shall be a general strategy for proving that a function is an invariant; in order to prove that a function is a link invariant, we prove its invariance under the Reidemeister moves.