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\begin{document}

\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Notice of Intent}
\vskip 2mm

I’m considered an expert on Knot Theory, yet I don’t understand knot
theory at all. From a certain perspective, Knot Theory is the study of
some silly combinatorial objects, that are considered modulo equally
silly relations.  My intuition as a student told me it must be a shallow
topic, and there’s still a remnant of that intuition in me.

Yet time after time this intuition is proven wrong and instead of shallow,
Knot Theory is very deep. So much so, that Knot Theory sometimes serves
to validate that other topics are interesting: {\bf if it has applications
to Knot Theory, it must be good.} (Historically, number theory’s raison
d'\^etre had been similar; recently cryptography became a further bonus).

My plan over the grant period would be to continue to use knot theory
as an excuse and as a benchmark to study several other topics, mostly
in algebra:

\begin{enumerate}

\item I plan to continue to study, along with Roland van der Veen and
others, how ``solvable approximation'' of semisimple Lie algebras
(Inonu-Wigner contractions of their lower Borel subalgebras) leads via
perturbed Gaussian formulas (in spirit, QFT) to poly-time computable
knot invariants that ``behave well'' under useful knot theoretic
operations. I hope this sounds powerful; it certainly sounds highly
technical. Can we make it less technical? Can we rely less on Lie algebra
and quantum algebra techniques and instead make the topic intrinsic to
knot theory? See \web{SolvApp}, \web{PG}, \web{DaNang}.

\item These invariants also have integral formulas, in terms of perturbed
Gaussian integrals, which reduce the proofs of their invariance to ``use
Fubini'' (see \web{ICBS}). Is there a direct knot theoretic reason to
expect such formulas?

\item The simplest of these invariants, $\rho_1$, is ridiculously simple
to define (\web{APAI}, \web{Cars}) and it is perhaps even more ridiculous
how much we fail to understand it. In short, $\rho_1$ is some quadratic
expression in the entries of $A^{-1}$, where $A$ is one of the standard
matrices whose determinant is the Alexander polynomial $\Delta$. Could
we start from other matrices $B$ whose determinants are $\Delta$? Can
we prove Alexander-like properties of $\rho_1$ using its similarity
with $\Delta$? By direct computations we observe many such properties,
yet we still don’t know how to prove them. And the \$1M question:
does $\rho_1$ have special properties on ribbon knots, similar to the
Fox-Milnor property of $\Delta$? If it does, it may lead to a new criteria
to detect non-ribbon knots. Such criteria are in high demand for they may
lead to the detection of counterexamples to the ribbon-slice conjecture,
one of the greatest outstanding problems in knot theory.

\item The second simplest of these invariants, $\theta$, is
presently the strongest genuinely-computable knot invariant known (see
\web{ICBS}). ``Genuinely computable'' means that we’ve computed it on
huge knots with over 250 crossings (a slight weakening can be computed
on knots with over 500 crossings). ``Strongest'' means that on the first
59,937 knots (up to 14 crossings) it attains 58,819 distinct values
(a deficit of 1,118), whereas the HOMFLY-PT polynomial and Khovanov
homology taken together (famous yet not as computable) have a much
greater deficit of 10,788. I plan to continue to study $\theta$.

\item Along with Zsuzsanna Dancso, Tamara Hogan, Jessica Liu, and Nancy
Scherich (\web{PDS}), I plan to continue to study knots and tangles in
a ``pole dancing studio'' (PDS, a cylinder with a few vertical lines
removed) and their relationship with the Goldman-Turaev Lie bialgebra
and Kashiwara-Vergne (KV) equations (\web{AKKN}). Are solutions of the KV
equations sufficient to construct a homomorphic expansion of tangles in
a PDS up to strand-strand degree 1?  How is this related to my earlier
work with Dancso (\web{WKO1}, \web{WKO2}) on welded knots? The subject
is beautiful, yet it is a hard-to-penetrate patchwork of results and
techniques and papers by different authors. In the past, this feeling
that a subject’s beauty is incongruous with its complexity had been
a great motivator for me, often leading to deeper understanding. I have
high hopes for this topic too.

\item Recently (\web{PQ}), along with Jessica Liu, we’ve found a truly
elegant ``signatures for tangles'' invariant (sorry for complimenting
ourselves, yet hey, it really is elegant). There is more to do before
we can claim to fully understand these signatures, and I hope to pursue
that over the grant period.

\end{enumerate}

\eject

\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal Summary}
\vskip 2mm

One of the major triumphs of mathematics in the 1980s, related to
at least 3 Fields medals (Jones, Drinfel'd, Witten) was the unexpected
realization that low dimensional topology, and in particular knot theory,
is closely related to quantum field theory and to the theory of quantum
groups. Knot theory is mundane and ages-old; anything ``quantum'' seems
hyper-modern. Why would the two have anything to do with each other?

The answer is long and complicated and has a lot to do with the
``Yang-Baxter Equation'' (YBE). The YBE on the one hand can be interpreted
in knot theory as ``the third Reidemeister move'', or as ``controlling
the most basic interaction of 3 pieces of string'' (this turns out to
be a very crucial part of knot theory). On the other hand solutions of
the YBE arise from ``quantum'' machinery. Hence the quantum is useful to
the knotted, and by similar ways, to the rest of low dimensional topology.

But ``quantum'' has a caveat, which makes it super-exciting (to some)
yet bounds its usefulness (to others). When quantum systems grow
large (as they do when the knot or low-dimensional space we study
grows complicated), their ``state space'' grows at an exponential
rate. ``Quantum computers'' aim to exploit this fact and make large
quantum systems performs overwhelmingly large computations by utilizing
their vast state spaces. But quantum computers aren't here yet, may take
many years to come, suffer from other limits on what they can do, and
much of low-dimensional topology is anyway outside of these limits. So
at least for now and likely forever, many things that have ``quantum''
in their description are exponentially-complex to compute, which in
practice means that they cannot be computed beyond a few simple cases.

Recently van der Veen and myself, following Rozansky and Overbay and
Ohtsuki, found a corner (figuratively speaking) of the vast state space
of the quantum machinery used in knot theory, which can be described
extremely simply, which computes in just polynomial complexity, and which
carries enough information to still speak to knot theory. The ``knot
invariants'' $\rho_d$ and $\theta$ constructed that way seem to be the strongest
invariants we know that are computable even for very large knots and
they have the potential of relating to knot properties such that their
genus and whether or not knots are slice or ribbon.

Our approach in itself comes from sophisticated quantum algebra, yet the
results can be described using nothing more than first-year university
mathematics. More often than not, when a result is simple there is also
a simple way to derive it, and it is often crucial to find that simple
way. We don't know yet how to tell the $\rho_d$ / $\theta$ story in a language as
simple as the formulas at the end of that story, and we dream that over
the grant period we will learn to do better.

We also dream to find topological applications of $\rho_d$ / $\theta$ and especially
of $\theta$, and to continue our work on other topics within knot theory.

\eject \setcounter{page}{1} \pagestyle{normal}

\par\noindent{\Large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal}
\vskip 2mm

\begin{multicols}{2}

{\bf Recent Progress.}

\if\draft y
{\em Describe your recent progress in research activities related to the proposal; for returning grantees, describe as well the progress attributable to your previous Discovery Grant.}
\fi

For the purpose of this proposal, I would like to concentrate on topic \#4 in my {\em Notice of Intent}. Recently, as a direct outcome of the research supported by my previous NSERC grant, Roland van der Veen and I commenced a study of a knot invariant $\Theta$, which is the {\bf strongest genuinely computable knot invariant presently known}. Let me start by discussing the words in this bold statement.

\parpic[r]{\includegraphics[width=0.5\linewidth]{figs/KnottedCandies@5x7_WB_800.png}}
A {\bf knot} is a piece of string tangled up in 3D space; some examples appear on the right. We consider two such tanglings to be equivalent if you can get from one to the other by continuously deforming the strings from one shape to the other, without cutting them at any point. Knots may appear esoteric, yet they are key to the understanding of all 3-dimensional and 4-dimensional spaces. For a light introduction, see~\cite{Muller:KnotTheory}.

It is in general very difficult to decide if two knots are equivalent; the best algorithms to do so take an exponential amount of time and hence they are impractical. So we seek what's called ``knot {\bf invariants}'' -- computable functions that assign to a knot some simpler quantities, such as polynomials or matrices, in a way so that equivalent knots are assigned equal invariants. It is even better if one can read topological properties of the knot from the values of its invariants.

By {\bf genuinely computable} we mean that we can compute $\Theta$ on arbitrary knots with up to about 300 crossings. For almost any other invariant presently known, that would be science fiction.

By {\bf strongest} we mean that $\Theta$ appears to be quite good at separating knots. For example, on the 313,230 prime knots with up to 15 crossings, $\Theta$ attains 306,472 distinct values -- a deficit of 6,758. The better known yet less computable HOMFLY-PT polynomial and Khovanov homology, taken together, have a deficit of 70,245, more than 10 times the worse.

%\parpic[r]{\includegraphics[width=0.6\linewidth]{figs/PP317.png}}
{\em Aside.} The main part of the value of $\Theta$ is a two-variable polynomial. Such polynomials can be regraded as 2D arrays of coefficients, and these can be considered as coding the colour values of pixels. Hence the values of $\Theta$ can be displayed as pictures. The picture corresponding to a random 317 crossing knot (from~\cite{DHOEBL:Random}) appears below. There are patterns in these pictures, and one of my less-major goals within the grant period will be to understand them.

\noindent\includegraphics[width=\linewidth]{figs/PP317.png}

\noindent\rule{\linewidth}{1pt}

While perhaps not strictly necessary, I'd like to give here a complete definition of $\Theta$.

%\needspace{30mm}
\parpic[r]{\input{figs/SampleDiagram.pdf_t}}
{\em Preparation.} Given an oriented knot $K$, we draw it in the plane
as a long knot diagram $D$ with $n$ crossings in such a way that the
two strands intersecting at each crossing are pointing up (that's always
possible because we can always rotate crossings as needed), and so that
at its beginning and at its end the knot is oriented upward. We label
each edge of the diagram with two integer labels: a running index $k$
which runs from 1 to $2n+1$, and a ``rotation number'' $\varphi_k$, the
geometric rotation number of that edge (the signed number of times the
tangent to the edge is horizontal and heading right, with cups counted
with $+1$ signs and caps with $-1$; this number is well defined because
at their ends, all edges are headed up). On the right the running index
runs from $1$ to $7$, and the rotation numbers for all edges are $0$
except for $\varphi_4$, which is $-1$.

{\em Making a matrix.} We let $A$ be the $(2n+1)\times(2n+1)$ matrix with
entries in the ring $\bbZ[T^{\pm 1}]$ of Laurent polynomials in a formal
variable $T$ obtained by starting with the identity matrix $I_{2n+1}$
and adding to it one contribution per crossing as follows ($s$ is the
sign of the crossing):

\begin{multline} \label{eq:Xings}
  \qquad\begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \qquad\longrightarrow \\
  \begin{array}{c|cccc}
    \text{add at} &   \text{column }\ip &  \text{column }\jp \\
    \hline
    \text{row }i & -T^s  & T^s-1 \\
    \text{row }j & 0  & -1
  \end{array}
\end{multline}

For our example, $A$ comes out to be:
\[
  A=\left(\begin{array}{ccccccc}
    1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\
    0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\
    0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\
    0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\
    0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\
    0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\
    0 & 0 & 0 & 0 & 0 & 0 & 1 \\
  \end{array}\right)
\]

Please count everything so far as ``trivial''. The matrix $A$ is a
presentation matrix for the Alexander module of $K$, obtained by using
Fox calculus on the lower Wirtinger presentation. Up to a unit $\pm
T^\bullet$, its determinant is the normalized Alexander polynomial
$\Delta$ and there's nothing new about it. In fact, 
\[ \Delta = T^{(-\varphi-w)/2}\det(A),
  \quad\text{with}\quad \varphi = \sum_k \varphi_k,\ w = \sum_c s.
\]
Note that in our example $\Delta = T-1+T^{-1}$.

{\em Doing something new}. Let $G = (g_{\alpha\beta}) = A^{-1}$ be the
inverse matrix of $A$, so in our example, $G$ is
\[ \left(\begin{array}{ccccccc}
   1 & T & 1 & T & 1 & T & 1 \\
   0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\
   0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\
   0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\
   0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\
   0 & 0 & 0 & 0 & 0 & 1 & 1 \\
   0 & 0 & 0 & 0 & 0 & 0 & 1 \\
  \end{array}\right).
\]

There is little precedence for inverting a presentation matrix, so already here we are in little-explored territory.

Let $T_1$ and $T_2$ be indeterminates and let $T_3=T_1T_2$. For $\nu=1,2,3$ let $\Delta_\nu$ and $G_\nu = (g_{\nu\alpha\beta})$ be $\Delta$ and $G$ subject to the substitution $T\to T_\nu$. Now define
\begin{multline} \label{eq:theta}
  \theta(K) \coloneqq \Delta_1\Delta_2\Delta_3\left(\sum_c R_{11}(c) + \sum_{c_0,c_1} R_{12}(c_0,c_1) \right. \\
  \left. + \sum_k\Gamma_1(\varphi_k,k)\right),
\end{multline}
where the first summation is over crossings $c=(s,i,j)$ (with $s$, $i$, $j$ as in~\eqref{eq:Xings}), the second is over pairs of crossings $(c_0=(s_0,i_0,j_0),c_1=(s_1,i_1,j_1))$, and the third is over edges $k$, and where
\[ \includegraphics[max width=\linewidth]{MathematicaInserts/R1Input.pdf} \]
\[ \includegraphics[max width=\linewidth]{MathematicaInserts/thetaInput.pdf} \]
and
\[ \includegraphics[max width=\linewidth]{MathematicaInserts/Gamma1Input.pdf} \]
(The formulas above were computer-generated from the source code of a program that computes $\Theta$ and that was verified throughly. This guarantees the absence of typos.)

Now let $\Theta(K)=(\Delta(K),\theta(K))$ (the computation of $\Delta$ is a part of the computation of $\theta$, so including it here is not artificial). This completes the definition of $\Theta(K)$. Yet, to emphasize that the definition above is actually quite simple, here is a complete implementation of $\Theta$, written in Matematica~\cite{Wolfram:Mathematica}:

\noindent\includegraphics[max width=\linewidth]{MathematicaInserts/ProgramInput.pdf}

\noindent{\bf Note 1.} We note the similarity between the formulas written here for $\Theta$ with evaluations of perturbed Gaussian integrals via Feynman diagrams. In both cases the end result is a sum of polynomials in the entries of the inverse of a matrix. This similarity can be made precise, and indeed, in~\cite{IType} I write a perturbed Gaussian integral formula for $\Theta$, in which the Lagrangian is a sum over the crossings of $K$ of quadratic terms that correspond to the matrix $A$ (and whose inverse becomes $G$) and of higher order perturbation terms. The integration is carried out over a space whose dimension is 6 times the number of edges in a diagram $D$ of $K$ --- a space that has some combinatorial significance (as it pertains to a knot diagram) but no immediate topological significance.

\noindent\rule{\linewidth}{1pt}

{\bf Objectives.}

\if\draft y
{\em Define the short- and long-term objectives of your research program. Note that a research program should have a long-term vision that expands beyond the five years of the Discovery Grant. A single, short-term project or collection of projects does not constitute a research program.}
\fi

I have two long-term career goals. They are radical and it is time to out them.

\noindent{\bf Goal 1.} {\em It's time to depreciate Witten-Reshetikhin-Turaev invariants (WRT) within knot theory and low dimensional topology.} This may sound like the words of a lunatic, seeing that so so much has been written about the Chern-Simons-Witten quantum field theory and about Reshetikhin-Turaev invariants (including by myself). Yet please, bear with me and keep an open mind:
\begin{itemize}
\item WRT invariants came to topology from outside, from representation theory and from quantum algebra and quantum field theory, and we still can't quite motivate them in the language of topology. Seen in the eyes of a topologist who studies objects (and not a dual-topologist, who studies invariants for their own sake), WRT invariants seem like artificial constructs.
\item Other than their separation power WRT invariants simply don't do much in topology.
\end{itemize}

These two bullets are not unrelated, of course. What isn't natural in topology is unlikely to do much for topology.

Yet topologists can't fairly pretend that WRT invariants don't exist. They exist for sure. We simply don't understand them.

One of my long-term career goals is to find the proper topology home for WRT invariants. I used to think that this entailed understanding them and the processes leading to them better, but it may be that by now I understand those well enough to suspect that that's not where the keys are hiding. I now believe in the following instead:

\begin{quote}
There is a natural home in topology for the invariant $\Theta$ discussed earlier, and in that home live many other invariants with formulas similar to $\Theta$'s. The collection of all such invariants is equivalent to the collection of WRT invariants; except that the WRT invariants make ``a wrong basis'' to that collection, within which it is hard to see their naturality and their utility.
\end{quote}

The matrix $g_{\alpha\beta}$, a key to the construction of $\Theta$, is the inverse matrix of a presentation matrix of the Alexander module $M$ of a knot $K$, the first homology of the universal Abelian cover of the knot complement. There are other presentation matrices for $M$ and I expect that many of them can be used to write alternative formulas for $\Theta$. In particular, I expect that there should be a ``Seifert formula'' for $\Theta$, presenting it as a perturbed Gaussian integral of the exponential of a Lagrangian $L$ which is a naturally-defined function on the homology $H$ of a Seifert surface $\Sigma$ for $K$ (or of a finite number of copies of that homology). The quadratic part of the Lagrangian should be the Seifert linking form, repeated over several copies of $H$ (and taken with different parameters $T_1$, $T_2$, \ldots).

Where would the Lagrangian $L$ be coming from? To compute the topologically most interesting knot polynomial, the Alexander polynomial, one only studies the linking and intersection numbers of curves on $\Sigma$. We are clearly missing a lot of topology here. I expect that other finite type invariants of curves on $\Sigma$ will be used to produce the ``perturbation'' terms of $L$.

In my dreams, given a knot $K$ we will pick a Seifert surface $\Sigma$ for $K$ with homology vector space $H$. We will then be asking ourselves, ``which Lagrangians $L$ on $nH$ ($n$ copies of $H$, for various fixed values of $n$), when integrated on $nH$ using the rules of perturbative Gaussian integration (namely, Feynman diagrams), will yield knot invariants''? I expect that the answer to that question is non-empty, for the currently strongest genuinely-computable knot invariant $\Theta$ is most likely an example (and a few more are at~\cite{IType}). We ought to be able to classify in simple terms the set of such Lagrangians. There should be plenty, and it should be possible to describe them in topologically more natural terms than ``semi-simple Lie algebras and their representations''. The corresponding invariants will be strong and easy to compute (as $\Theta$ is) and they should be topologically meaningful, as they will be intrinsically aware of the topology of the Seifert surface $\Sigma$.

I expect that ultimately these Seifert-type invariants will replace WRT invariants as a center of attention in algebraic knot theory and low dimensional topology.

\noindent{\bf Goal 2.} {\em Disprove the ribbon-slice conjecture}. It is well known that every ribbon knot is slice, and one of the greatest open problems in knot theory is whether the converse holds true; namely, whether every slice knot is ribbon. This possible equivalence of ribbon knots with slice knots is known as the ``ribbon-slice conjecture'', and many people believe it to be false. There are even proposed counterexample (e.g., in~\cite{GompfScharlemannThompson:Counterexample}, and note that these proposed counterexamples are rather large knots). What is missing is a proof that these proposed counter examples aren't ribbon knots.

What is needed is a knot invariant $\Psi$ whose values on ribbon knots are especially constrained. If such an invariant can be computed on those rather large proposed counterexamples, and if its values on the proposed counterexamples do not satisfy the ribbon constraints, we will have disproved the ribbon-slice conjecture.

So we need an invariant $\Psi$ that can be computed efficiently on large knots, and that ``sees'' the Seifert surface of a knot (as the Seifert surfaces of ribbon knots can be taken to be of a special form, possibly leading to the restrictions on the values of $\Psi$; see ``Seifert for Ribbon'' below).

If there ever was a good candidate for $\Psi$, it is our $\Theta$ (in fact, a lot of my motivation for the development of $\Theta$ was precisely that it would serve as $\Psi$). Yet a lot of work remains to be done. I am sure there are Seifert formulas for $\Theta$, but I don't know them yet. And once these formulas are written, it would still be necessary to find what constraints on their values can be obtained from the existence of Seifert surfaces of the form that arises from ribbons. This is a hard problem and I expect it will take several years to find the answer. 

{\em Seifert for Ribbon.} A ribbon knot with $g$ ribbon singularities always has a Seifert surface $\Sigma$ of genus $g$, in which $g$ of the $2g$ homology cycles can be jointly represented by a $g$-component unlink. See Figure~1.

If as I believe $\Theta$ has a Seifert formula as discussed in Goal 1, in which all the ingredients of the Lagrangian $L$ are finite type invariants of curves representing homology classes on $\Sigma$, the half-triviality of these curves as indicated above will lead to strong restrictions on the form of $L$ which in themselves may lead to strong restrictions on the values of $\Theta$.

For the Alexander polynomial $\Delta$, the same reasoning leads to the Fox-Milnor condition. See e.g.~\cite[pp. 212-213]{Kauffman:OnKnots}.

\begin{Figure}
\[ \mathrlap{(a)}\includegraphics[width=0.75\linewidth]{figs/RibbonKnot.pdf} \]
\[
  \mathrlap{(b)}\includegraphics[width=0.5\linewidth]{figs/RibbonSingularity.pdf}
  \mathrlap{(c)}\includegraphics[width=0.5\linewidth]{figs/Seifert4Ribbon.pdf}
\]
\captionof{figure}{\footnotesize A ribbon knot $(a)$ and a ribbon singularity $(b)$ (singularities in green), and a piece of a Seifert surface for a ribbon knot near a ribbon singularity $(c)$, with an unknotted homology cycle in green. A ribbon knot with $g$ ribbon singularities will have $g$ of those, unlinked with each other.}
\end{Figure}

As for the other topics within my Notice of Intent (\web{NOI}): Topic \#4 is the core of this proposal as above.
Topic \#2 was mentioned in passing within the above, and will not be mentioned
further. Topic \#3 is mostly subsumed within the discussion of topic \#4 above. To a large extent, $\rho_1$ is very much like $\theta$ except with somewhat different specific formulas. For topics \#1, \#5, and \#6, I will simply repeat \web{NOI} with
some modifications:

\noindent{\bf Topic 1.} I plan to continue to study, along with Roland van der Veen and
others, how ``solvable approximation'' of semisimple Lie algebras
(Inonu-Wigner contractions of their lower Borel subalgebras) leads via
perturbed Gaussian formulas (in spirit, QFT) to poly-time computable
knot invariants that ``behave well'' under useful knot theoretic
operations. I hope this sounds powerful; it certainly sounds highly
technical. Can we make it less technical? Can we rely less on Lie algebra
and quantum algebra techniques and instead make the topic intrinsic to
knot theory? See \cite{SolvApp, PP1, PG, DoPeGDO}.

\noindent{\bf Topic 5.}
Along with Zsuzsanna Dancso, Tamara Hogan, Jessica Liu, and Nancy
Scherich~\cite{PDS}, and also along with Yusuke Kuno~\cite{EmergentAssociator},
I plan to continue to study knots and tangles in
a ``Pole Dancing Studio'' (PDS, a cylinder with a few vertical lines
removed) and their relationship with the Goldman-Turaev Lie bialgebra
and Kashiwara-Vergne (KV) equations~\cite{AKKN1, AKKN2}. Are solutions
of the KV equations sufficient to construct a homomorphic expansion of
tangles in a PDS up to strand-strand degree 1? How is this related to my
earlier work with Dancso~\cite{WKO1, WKO2} on welded knots? The subject
is beautiful, yet it is a hard-to-penetrate patchwork of results and
techniques and papers by different authors. In the past, this feeling
that a subject's beauty is incongruous with its complexity had been a
great motivator for me, often leading to deeper understanding. I have
high hopes for this topic too.

\noindent{\bf Topic 6.}
Recently~\cite{PQ}, along with my Ph.D.\ student Jessica Liu, we've found a truly elegant
``signatures for tangles'' invariant (sorry for complimenting ourselves,
yet hey, it really is elegant). There is more to do before we can claim
to fully understand these signatures. Is there an Alexander invariant
for tangles obtained using the same ``pushforward'' techniques? Are
its roots related to the jumping points of the signature? Does
it generalize to the multi-variable case? This topic was originally conceived within an attempt to prove the  Kashaev Signature
Conjecture~\cite{Kashaev:SymmetricMatrices}, but that conjecture is by
now my student's Jessica Liu's theorem~\cite{Liu:ProofOfKashaev}.

\needspace{20mm}
{\bf Literature review.}

\if\draft y
{\em Discuss the literature pertinent to the proposal, placing the proposed research in the context of the state of the art.}
\fi

$\Theta$ is most likely equal to ``the two loop contribution to the Kontsevich integral'', as studied by Garoufalidis, Rozansky, Kricker, and in great detail by Ohtsuki \cite{GaroufalidisRozansky:LoopExpansion, Ro, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines, Ohtsuki:TwoLoop}, continuing an older study by myself and Garoufalidis~\cite{MMR} (we haven't proven that, yet I expect we will soon). But the definitions used by these authors are a lot more complicated than ours, and do not lend themselves to efficient computations.

$\Theta$ is probably related to the invariant considered by Garoufalidis, Kashaev, and Li in~\cite{GaroufalidisKashaev:Multivariable, GaroufalidisLi:Patterns}, as they share many properties. Yet they are different, and our definitions are simpler and lead to vastly faster computations.

Other than that, there isn't a lot written yet about $\Theta$.

%\if\draft y \newpage \fi
{\bf Methodology.}

\if\draft y
{\em Describe the methods and proposed approach, providing sufficient details to allow the reviewers to assess the feasibility of the research activities.

Considering equity, diversity and inclusion (EDI) in the research process promotes research excellence by making research outcomes more ethically sound, rigorous, reproducible, and useful. It is important to consider EDI through each stage of the research process, including, but not limited to, the research questions, design, methodology, analysis, interpretation and dissemination of results, and integrate these considerations where relevant. Consult Equity, diversity and inclusion considerations at each stage of the research process for more information.}
\fi

To some extent my methodology is as dull as it gets. I sit in my office (with my feet up if nobody's looking), or in a coffee shop (never with my feet up), or I ride the bus or I lie in bed, and I think. Sometimes a pencil and a piece of paper are involved too.

Yet in one way my methodology differs from that of most mathematicians. Almost everything I do I implement on the computer almost immediately. It isn't just that I implement what I conceived with pencil and paper once the latter matures. Rather, the pencil and paper and the implementation are fully interleaved and integrated. My implementations are mathematically-informed, and my thoughts and scribbles never diverge much from what can be implemented. I believe that $\Theta$, the strongest genuinely-computable knot invariant we presently know, is a great success. Its strength is the result of a bit of informed luck. Its computability isn't luck. Its computability is because I think about computability at nearly all times. Computability was a part of the development process of $\Theta$ throughout. I expect that my future work will follow the same lines.

I am asked to comment here on issues of EDI (Equity, Diversity, and Inclusion). On the surface, such issues do not arise in mathematical research or within my methodology of research. Math is gender- and race-neutral, and computers don't know the races and genders of the people punching their keyboards. This said, I am aware, and over the years I became more and more aware, of how differing backgrounds may lead to differing levels of initial preparedness, of how differing societal and cultural expectations lead to different ways in which we present ourselves, and of how inconsiderate feedback, or worse, the wrong kind of attention, can greatly harm the motivation and success of young researchers.

With this (ever growing) awareness I'm doing my best to make the atmosphere in my research group supportive to all members of all under-represented groups. The alternative, of losing great minds because perhaps they dress or look differently, would be offensively stupid.

I think I've had some success. Of my 17 PhD students so far, 5 are women: one is a current student, and the four that have already graduated all continued within academia, two with tenured or tenure-track positions (at the University of Sydney and at Northeastern University). My most recent graduated PhD student, who defended his thesis last August, came from Ghana to Canada back in 2017 specifically in order to work with me, following links I have established when I volunteered to give a course on algebraic topology at the University of Ghana in 2010. I've had (and I have) a number of other BIPOC students, but the definitions here are sometimes ambiguous and do not belong in this document, so I will refrain from including statistics. Of my 5 post-doctoral fellows, 3 were women (one is current). 

%\if\draft y \newpage \fi
{\bf Impact.}

\if\draft y
{\em Explain the anticipated significance of the work.}
\fi

I expect the work proposed here to revolutionize what we know about knot invariants. The invariant $\Theta$ is already the strongest genuinely-computable invariant we have, and it stands to get better by acquiring a solid topological foundation. In addition, I think there is a fair chance that the work that I propose will lead to disproving the ribbon-slice conjecture, one of the most major outstanding problems in knot theory.

\end{multicols}

\eject \pagestyle{empty}

\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal Budget Justification}
\vskip 2mm

\begin{multicols}{2}

{\bf Salaries and Benefits.} Since 2017 I have graduated four PhD
students (Travis Ens, Jesse Frohlich, Huan Vo, Leonard Afeke). I am presently working
with three more (Jessica Liu, Daniel Martchenkov, Kevin Santos). I
plan to support each of those at around \$10,000 per year. In addition
I've had a number of master's students, I expect to have about two more
per year, and to support each at about \$5,000 per year. Likewise I've
taken a number of undergraduate ``summer project'' students, and I hope
to support about two such students per year, at about \$2,500 each.

I hope to be able to support a postdoctoral fellow throughout the grant
period, at about \$50,000 per year.

{\bf Equipment or Facility.} Many of my past projects required massive
computations, often running for months at a time (e.g., the calculation of
all the invariants appearing on the Knot Atlas, \url{http://katlas.org}),
and many of the results are made available by means of a dedicated web
server, \url{http://drorbn.net}, especially \url{http://drorbn.net/ap}. My
current proposal will lead me to continue using computers in a similar
way. This will be a lot more effective if I would be able to purchase
and maintain current hardware. Hence the \$3,200 allocated per year for
purchase or rental of computers and peripherals, and the \$700 allocated
per year for the maintenance of those. Also, I will have to pay user
fees for some of the programs I will be using (Mathematica, for example)
and also to some shared facilities to be provided by my university ---
internet connection, backup services, etc. I am requesting an amount of
\$2,000 per year for these purposes.

{\bf Materials and Supplies.} This amount of \$700 per year will be used
primarily to purchase office supplies and printer paper and ink.

{\bf Travel.} In the past I have traveled extensively and gave
presentations on my work in a large number of domestic and foreign
universities and in many international conferences. I expect this will
continue throughout the years of my contract. In addition I hope to
support some travel by my graduate students and postdoctoral fellows,
and to support visits by my scientific collaborators to Toronto. I am
requesting an amount of \$9,000 per year for these purposes.

{\bf Books.} Need no explain.

\end{multicols}

\eject \pagestyle{empty}

\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal HQP Training Plan}
\vskip 2mm

My project clearly spreads in several directions. This means that there is
ample room for advanced undergraduate students, for graduate students,
and for postdoctoral fellows to take part in the research outlined
in my proposal and/or in closely related research. This allowed me to
participate in the training of many students and fellows in the past,
and will continue to allow me to do the same in the future.

I share my significant use of computers as a tool for research,
presentation and dissemination of knowledge with my students and
postdoctoral fellows. I believe this  adds major further quality to the
training they receive.

\vskip 5mm
Though frankly, I still don't know how to do the thing I'd really want
to do.

For me, the best mathematics is the math that can be implemented on a
computer. This ranges from the simplest, say Gaussian elimination or the
Fibonacci sequence, and continues all the way to the fanciest and most
abstract, be it a planar-algebra category-theory ultra-fast computation
of Khovanov homology or a free-Lie-algebra meta-group-action-based
computation of a non-commutative generalization of the Alexander
polynomial or the implementation of the full portfolio of operations
around the quantum universal enveloping algebras of solvable
approximations of semisimple Lie algebras. I've implemented these,
as well as a dozen other versions of the Alexander polynomial, and a
dozen other knot invariants, and a very large number of other little
things within knot theory, and a computer solution of the Rubik's cube,
and a hyperbolic-geometry based algorithm for optimal camera motion, and
I made computer generated pictures of various fancy links and surfaces
and of steps within Arnold's resolution of Hilbert's 13th problem,
and very many other things, big and small. (And most are on my web site).

For me, that's what keeps mathematics alive and sincere and believable
(and when it comes to the graphics, sometimes also visually beautiful).

I wish I knew how to teach my students to actually compute (and draw)
what they are talking about, and gain the benefit that that entails, and
pass it on to their students later on. I wish they would do it routinely
and often, and with joy. I think I've contributed some, and I hope to
contribute further, to my students by sharing with them my love of the
implementable (and teaching them a bit of the how-to).

A side benefit is that if and when my students do become proficient with implementation, they become highly desirable in plenty of other fields of science and industry.

\vfill

\par\noindent{\bf Knot Theory as an Excuse \hfill Discovery Grant Relationship to Other Research Support}
\vskip 2mm

I am the lucky recipient of a C\$248,447 grant from the Chu Family
Foundation (NYC), used to fully fund a post-doctoral fellow, Tamara Hogan, for a period of 3 years. Dr.\ Hogan is working on what amounts to Topic \#5 in my proposal. This grant cannot be used for any
other purpose.

\vfill

\eject

\par\noindent{\bf Knot Theory as an Excuse \hfill Discovery Grant Most Significant Contributions}
\vskip 2mm

{\bf Computing Finite Type Invariants Efficiently,} with I.~Bar-Natan, I.~Halacheva, and N.~Scherich, \linebreak[4]\arXiv{2408.15942}.

\noindent{\em Abstract.} We describe an efficient algorithm to compute finite type invariants of type $k$ by first creating, for a given knot $K$ with $n$ crossings, a look-up table for all subdiagrams of $K$ of size $\lceil\frac{k}{2}\rceil$ indexed by dyadic intervals in $[0,2n-1]$. Using this algorithm, any such finite type invariant can be computed on an $n$-crossing knot in time $\sim n^{\lceil\frac{k}{2}\rceil}$, a lot faster than the previously best published bound of $\sim n^k$.

{\bf A Perturbed-Alexander Invariant,} with R.~van der Veen, {\em Quantum Topology} {\bf 15} (2024) 449--472, \web{APAI}, \arXiv{2206.12298}.

\noindent{\em Abstract.} In this note we give concise formulas, which
lead to a simple and fast computer program that computes a powerful
knot invariant. This invariant $\rho_1$ is not new, yet our formulas
are by far the simplest and fastest: given a knot we write one of the
standard matrices $A$ whose determinant is its Alexander polynomial,
yet instead of computing the determinant we consider a certain quadratic
expression in the entries of $A^{-1}$. The proximity of our formulas
to the Alexander polynomial suggest that they should have a topological
explanation. This we don't have yet.

{\bf Perturbed Gaussian Generating Functions for Universal Knot
Invariants,} with R.~van der Veen, \linebreak[4]\arXiv{2109.02057}.

\noindent{\em Abstract.} We introduce a new approach to universal quantum
knot invariants that emphasizes generating functions instead of generators
and relations. All the relevant generating functions are shown to be
perturbed Gaussians of the form $Pe^G$, where $G$ is quadratic and $P$
is a suitably restricted ``perturbation''. After developing a calculus for
such Gaussians in general we focus on the rank one invariant $Z_\bbD$ in
detail. We discuss how it dominates the $sl_2$-colored Jones polynomials
and relates to knot genus and Whitehead doubling. In addition to being
a strong knot invariant that behaves well under natural operations on
tangles $Z_\bbD$ is also computable in polynomial time in the crossing
number of the knot. We provide a full implementation of the invariant
and provide a table in an appendix.

{\bf Over then Under Tangles,} with Z.~Dancso and R.~van der Veen,
{\em Journal of Knot Theory and its Ramifications} {\bf 32-8} (2023),
\arXiv{2007.09828}.

\noindent{\em Abstract.} Over-then-Under (OU) tangles are oriented tangles whose strands travel
through all of their over crossings before any under crossings. In this
paper we discuss the idea of {\em gliding}: an algorithm by which
tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain
a braid classification result, which we also extend to virtual braids,
and provide a Mathematica implementation. We discuss other instances
of successful ``gliding ideas'' in the literature --
sometimes in disguise -- such as the Drinfel'd double construction,
Enriquez's work on quantization of Lie bialgebras, and Audoux and
Meilhan's classification of welded homotopy links.

{\bf Handout Portfolio.} I see lecturing and the assimilation of
mathematical knowledge and the exposition of its beauty as one of
my primary goals. I aim to polish my lectures to perfection; almost
every lecture I give comes with a colourful handout summarizing the
information in it, and with a web space with links to said handout, to
relevant papers and programs, and almost always, with a link to a video
recording of the talk itself. My 5th attached contribution is merely a
reminder of that --- an abridged version of my {\bf Handout Portfolio}
(the full version is at \url{http://drorbn.net/hp}).

\vfill

\par\noindent{\bf Knot Theory as an Excuse \hfill Discovery Grant 4 Samples of Research Contributions}
\vskip 2mm

See \web{Rooting}, \web{APAI}, \web{PG}, \web{OU}, and \web{hp}.

\vfill

\eject \setcounter{page}{1} \pagestyle{normal}

\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal References}
\vskip 2mm

\begin{multicols}{2}

\input{refs.tex}

\end{multicols}

%\eject
%
%\par\noindent{\large\bf Knot Theory as an Excuse \hfill Discovery Grant Proposal To Do List}
%\vskip 2mm
%
%\begin{itemize}
%\item Attach CCV.
%\item Reread instructions pages.
%\item Activate all the links.
%\item Clear this list and remove this page.
%\end{itemize}

\end{document}
