\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amscd,dbnsymb,graphicx,stmaryrd,picins,fancyhdr}
\usepackage{txfonts}    % for the likes of \coloneqq.
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
% Following http://tex.stackexchange.com/questions/69901/how-to-typeset-greek-letters:
\usepackage[LGR,T1]{fontenc}
\newcommand{\textgreek}[1]{\begingroup\fontencoding{LGR}\selectfont#1\endgroup}
\usepackage[all]{xy}
\usepackage[paperwidth=8.5in,paperheight=11in,textwidth=8.5in,textheight=11in]{geometry}
\usepackage{pdfpages}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\pensieve{http://drorbn.net/AcademicPensieve}

%\paperwidth 8.5in
%\paperheight 11in
%\textwidth 7in
%\textheight 9.5in
%\oddsidemargin -0.25in
%\evensidemargin \oddsidemargin
%\topmargin -0.25in
%\headheight 0in
%\headsep 0in
%\footskip 0.25in
%%\parindent 0in
%\setlength{\topsep}{0pt}

\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\calA{{\mathcal A}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\fraka{{\mathfrak a}}
\def\frakg{{\mathfrak g}}

\def\AS{\mathit{AS}}
\def\gr{\operatorname{gr}}
\def\IHX{\mathit{IHX}}
\def\STU{\mathit{STU}}
\def\TC{\mathit{TC}}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}}
\def\web#1{{\href{http://drorbn.net/K17/#1}{\textgreek{web}/#1}}}

\renewcommand{\headrulewidth}{0pt}
\fancypagestyle{first}{
  \lhead{\footnotesize\href{\myurl}{Dror Bar-Natan}: \href{\pensieve}{Academic Pensieve}: \href{\pensieve/Projects}{Projects}: \href{http://drorbn.net/K17}{Killam-2017}:}
  \rhead{\footnotesize\textgreek{web}$\coloneqq$\url{http://drorbn.net/K17/}}
}
\fancypagestyle{rest}{
  \lhead{\footnotesize\href{\myurl}{Dror Bar-Natan}}
  \rhead{\footnotesize\textgreek{web}$\coloneqq$\url{http://drorbn.net/K17/}}
}
\pagestyle{empty}

\def\red{\color{red}}

\catcode`\@=11
\long\def\@makecaption#1#2{%
    \vskip 10pt
    \setbox\@tempboxa\hbox{%\ifvoid\tinybox\else\box\tinybox\fi
      \small\sf{\bfcaptionfont #1. }\ignorespaces #2}%
    \ifdim \wd\@tempboxa >\captionwidth {%
        \rightskip=\@captionmargin\leftskip=\@captionmargin
        \unhbox\@tempboxa\par}%
      \else
        \hbox to\hsize{\hfil\box\@tempboxa\hfil}%
    \fi}
\font\bfcaptionfont=cmssbx10 scaled \magstephalf
\newdimen\@captionmargin\@captionmargin=2\parindent
\newdimen\captionwidth\captionwidth=\hsize
\catcode`\@=12

\begin{document} \thispagestyle{empty}


\parpic[r]{\parbox{1.25in}{
  \vskip -3mm
  \[ \input{ExampleT.pdf_t} \]
  \vskip -5mm
  \caption{A tangle.} \label{fig:Tangle}
}}
\noindent{\bf Second Answer.} The second answer has to do with ``Algebraic Knot Theory'', so let me start with that. Somewhat informally, a ``tangle'' is a piece of a knot, or a ``knot with endpoints'' (an example is on the right). Knots can be assembled by stitching together the strands of several tangles, or the different strands of a single tangle. Some interesting classes of knots can be defined algebraically using tangles and these stitching operations. Here is the most interesting example:

\noindent{\bf Definition 1.} A ``ribbon knot'' is a knot $K$ that can be presented as the boundary of a disk $D$ which is allowed to have ``ribbon singularities'' but not ``clasp singularities''. See Figure~2.

\vskip -4mm
\begin{figure}[h]
\[
  \includegraphics[height=18mm]{../../2015-07/PolyPoly/RibbonSingularity.png}
  \qquad
  \includegraphics[height=18mm]{../../2015-07/PolyPoly/ClaspSingularity.png}
  \qquad
  \includegraphics[height=24mm]{RibbonKnot.png}
  \parbox[b]{3in}{\setlength\captionwidth{3in}
    \caption{A ribbon singularity, a clasp singularity, and an example of a ribbon knot.}
  }\label{fig:Ribbon}
\]
\end{figure}
\vskip -6mm

\parpic[r]{\parbox{4.15in}{\vskip -3mm
\[ \input{TauKappa.pdf_t} \]
\[\text{gives}\quad
  \xymatrix{\calT_n & \calT_{2n} \ar[l]_\tau \ar[r]^\kappa & \calT_1}
\]
%\caption{The definitions of $\tau$ and $\kappa$.}\label{fig:TauKappa}
}}
\noindent{\bf Definition 2.} Let $\calT_{2n}$ denote the set of all tangles $T$ with $2n$ components that connect $2n$ points along a ``top end'' with $2n$ points along a ``bottom end'' inducing the identity permutation of ends (an example is the tangle in Figure~\ref{fig:Tangle}). Given $T\in\calT_{2n}$, let $\tau(T)$ be the result of stitching its components at the top in pairs as on the right --- it is an $n$-component tangle all of whose ends are at the bottom, and we (somewhat loosely) denote the set of all such by $\calT_n$, so $\tau\colon\calT_{2n}\to\calT_n$. Likewise let $\kappa(T)$ be the result of stitching $T$ both at the top and at the bottom, also as on the right. So $\kappa(T)$ is a 1-component tangle, which is the same as a knot, and $\kappa\colon\calT_{2n}\to\calT_1$.

\noindent{\bf Theorem 1} (I have not seen this theorem in the literature, yet it is not difficult to prove). The set of ribbon knots is the set of all knots $K$ that can be written as $K=\kappa(T)$ for some tangle $T$ for which $\tau(T)$ is the untangled (crossingless) tangle $U$:
\[ \{\text{ribbon knots}\} = \left\{\kappa(T)\colon T\in\calT_{2n}\text{ and }\tau(T)=U\in\calT_n\right\}. \]

\parpic[r]{\parbox{1.9in}{\vskip -3mm
\begin{equation} \label{eq:AKTDiagram} \xymatrix{
  \calT_n \ar[d]^Z
  & \calT_{2n} \ar[l]_\tau \ar[r]^\kappa \ar[d]^Z
  & \calT_1 \ar[d]^Z
  \\
  A_n
  & A_{2n} \ar[l]_{\tau_A} \ar[r]^{\kappa_A}
  & A_1
}\end{equation}
}}
Now suppose we have an invariant $Z\colon\calT_k\to A_k$ of tangles, which takes values in some spaces $A_k$. Suppose also we have operations $\tau_A\colon A_{2n}\to A_n$ and $\kappa_A\colon A_{2n}\to A_1$ such that the diagram on the right is commutative. Then
\begin{equation} \label{eq:RA}
  Z(\{\text{ribbon knots}\})\subseteq\calR_A\coloneqq\left\{\kappa_A(\zeta)\colon \zeta\in A_{2n}\text{ and }\tau_A(\zeta)=1_A\in A_n\right\}\subset\calA_1,
\end{equation}
where $1_A\coloneqq Z(U)\in A_n$. If the target spaces $A_k$ are algebraic (polynomials, matrices, matrices of polynomials, etc.) and the operations $\tau_A$ and $\kappa_A$ are algebraic maps between them (at this stage, meaning just ``have simple algebraic formulas''), then $\calR_A$ is an algebraically defined set. Hence we potentially have an algebraic way to detect non-ribbon knots: if $Z(K)\notin\calR_A$, then $K$ is not ribbon.

\parpic[r]{\includegraphics[height=1.3in]{GST48.pdf}}
As it turns out, it is valuable to detect non-ribbon knots. Indeed the Slice-Ribbon Conjecture (Fox, 1960s) asserts that every slice knot (a knot in $S^3$ that can be presented as the boundary of a disk embedded in $B^4$) is ribbon. Gompf, Scharlemann, and Thompson~\cite{GompfScharlemannThompson:Counterexample} describe a family of slice knots which they conjecture are not ribbon (the simplest of those is on the right). With the algebraic technology described above it may be possible to show that the~\cite{GompfScharlemannThompson:Counterexample} knots are indeed non-ribbon, thus disproving the Slice-Ribbon Conjecture.

What would it take?

\begin{itemize}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\item[C1.] An invariant $Z$ which makes sense on tangles and for which diagram~\eqref{eq:AKTDiagram} commutes.
\item[C2.] $Z$ cannot be a simple extension of the Alexander polynomial to tangles, for by Fox-Milnor~\cite{FoxMilnor:Singularities} the Alexander polynomial does not detect non-ribbon slice knots.
\item[C3.] $Z$ cannot be computable from finitely many finite type invariants, for this would contradict the results of Ng~\cite{Ng:Ribbon}.\footnote{A slight subtlety arises: There is no taking limits here, and C3 does not preclude the possibility that $Z$ is computable from infinitely many finite type invariants. The Fox-Milnor condition on the Alexander polynomial of ribbon knots, for example, is expressible in terms of the full Alexander polynomial, yet not in terms of any finite type reduction thereof. Unfortunately by C2 it cannot be used here.}
\item[C4.] $Z$ must be computable on at least the simplest~\cite{GompfScharlemannThompson:Counterexample} knot, which has 48 crossings.
\item[C5.] It is better if in some meaningful sense the size of the spaces $A_k$ grows slowly in $k$. Indeed in~\eqref{eq:RA}, if $A_{2n}$ is much bigger than $A_n$ and $A_1$ then at least generically $\calR_A$ will be the full set $A_1$ and our condition will be empty.
\end{itemize}

No invariant that I know now meets these criteria. Alexander and Vassiliev fail C2 and C3, respectively. Almost all quantum invariants and knot homologies pass C1-C3, but fail C4. Jones, HOMFLY-PT and Khovanov potentially pass C4, yet fail C5. We must come up with something new.

\renewcommand{\section}[2]{}%
\begin{thebibliography}{GST}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\bibitem[FM]{FoxMilnor:Singularities} R.~H.~Fox and J.~W.~Milnor,
  {\em Singularities of 2-Spheres in 4-Space and Cobordism of Knots,}
  Osaka J.\ Math.\ {\bf 3} (1966) 257--267.

\bibitem[GST]{GompfScharlemannThompson:Counterexample} R.~E.~Gompf,
  M.~Scharlemann, and A.~Thompson,
  {\em Fibered Knots and Potential Counterexamples to the Property 2R and
    Slice-Ribbon Conjectures,}
  Geom.\ and Top.\ {\bf 14} (2010) 2305--2347, \arXiv{1103.1601}.

\bibitem[Ng]{Ng:Ribbon} K.~Y.~Ng,
  {\em Groups of ribbon knots,}
  Topology {\bf 37} (1998) 441--458, \arXiv{q-alg/9502017} (with an
  addendum at \arXiv{math.GT/0310074})

\end{thebibliography}

\end{document}