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\begin{document}

\frontmatter

\title{Atlas of Tight Knots and Links}
\date{Revised: \today}

\author{Ted Ashton}
\email{ashted@ashtonfam.org}
\address{}

\author{Jason Cantarella}
\email{cantarel@math.uga.edu}
\address{Department of Mathematics, University of Georgia, Athens, GA 30602}

\author{Michael Piatek}
\email{piatek@cs.washington.edu}
\address{Department of Computer Science, University of Washington,Seattle, WA  98195}

\author{Eric Rawdon}
\email{rawdon@mathcs.duq.edu}
\address{Department of Mathematics, University of St. Thomas,St. Paul, MN}

\maketitle

\mainmatter

\chapter{Introduction}

This book compiles all of our currently known data about tight shapes of knots and links into a single source. We include tables of upper bounds for ropelengths of knots and links, some information about the relationship between ropelength and crossing number, and pictures of the tight shapes including their self-contact sets and curvature graphs. The result is an overview of close to 400 different knot and link types.

\section{Table of polygonal ropelengths and ropelength upper bounds}

Tables~\ref{ByKnotTableone},\ref{ByKnotTabletwo}, and \ref{ByKnotTablethree} show new polygonal ropelengths and ropelength upper bounds for $384$ knots and links with $10$ or fewer crossings. From left to right, there are repeating sets of three columns: the name of the knot or link, the polygonal ropelength $\PRop$ and the corresponding ropelength upper bound $\Rop$. 

\input{KnotTable.tex}

\section{Table of Knots and Links in Ropelength Order}

Table~\ref{ByRopTableone} shows our list of prime knots and links ordered by ropelength. This table is to be read vertically down each column from the left column to the right column. Notice that we have some data for 10 crossing knots, but no data for 10 crossing links. Further, we observe that ropelength is certainly not monotonic in crossing number. This means that the table may be somewhat misleading, since we expect 10 crossing links, and some knots and links of higher crossing number, to be interspersed with the data we present.

\section{Images of Tight Knots and Links}

Pages~\pageref{TwoTwoOnePage}-\pageref{TenOneHundredAndSixtySixPage} of the \emph{Atlas} contain images of the approximately tight configurations that we have computed and their strut sets. The pictures also include some additional information identifying geometric structures of interest in the tight curves. The graph along the bottom of the picture shows our numerical approximation of the curvature of the computed curve. When the curvature reaches its maximum value, these data points of maximum curvature are highlighted with red lines. These points are called ``kinks'', and the number of kinked intervals of the curve is counted on the data table under the heading ``kinks''. An example is shown in the $3_1$ knot on page~\pageref{ThreeOnePage}. On the right hand side, we give the arclength position of each kink on the file provided in the accompanying distribution.

We also highlight in blue regions of the curve which have no struts or kinks. According to theory, these should be straight line segments. An example is shown on page~\pageref{SixThreeThreePage}. Again, on the right hand side we see the arclength coordinates of the start and end of the strut-free region.

\input{KnotTableByRop.tex}

\newpage

\include{Bestiary}

\backmatter

\printindex

\end{document}