\documentclass[11pt]{article} \usepackage{amsmath,amssymb,amscd,dbnsymb,graphicx,stmaryrd,picins,fancyhdr,import, bbm, multicol} \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[textwidth=7in,textheight=9.5in,headsep=0.15in]{geometry} \usepackage{pdfpages} \def\myurl{http://www.math.toronto.edu/~drorbn} \def\pensieve{http://drorbn.net/AcademicPensieve} \def\bbe{\mathbbm{e}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calF{{\mathcal F}} \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\frakb{{\mathfrak b}} \def\frakh{{\mathfrak h}} \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/d17/#1}{\textgreek{web}/#1}}} \renewcommand{\headrulewidth}{0pt} \fancypagestyle{first}{ \rhead{\footnotesize\href{\myurl}{Dror Bar-Natan}} \lhead{\footnotesize\textgreek{web}$\coloneqq$\url{http://drorbn.net/d17/}} } \fancypagestyle{rest}{ \rhead{\footnotesize\href{\myurl}{Dror Bar-Natan}} \lhead{\footnotesize\textgreek{web}$\coloneqq$\url{http://drorbn.net/d17/}} } \pagestyle{rest} \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{first} \par\noindent{\large\bf Poly-Time Knot Theory and Quantum Algebra \hfill Discovery Grant Notice of Intent} \vskip 2mm {\small Totally by definition, once in a lifetime, a researcher is working on his personal best project. For me this is now, and I'm very excited about it. Let me explain. Here and there math has immense philosophical value or beauty to justify the effort. Yet everyday math is mostly about, or should be about, ``doing useful things''. Deciding if $A$ has property $B$, counting how many $C$'s satisfy $D$, computing $E$. When $A$ and $B$ and $C$ and $D$ and $E$ are small, we do the computations on the back of an envelope and write them as ``Example 3.14'' in some paper. But these are merely the demos, and sooner or later we worry (or ought to worry) about bigger inputs. I'm more aware then most mathematicians (though perhaps less than many computer scientists), how much the complexity of obtaining the solution as a function of the size of the inputs matters. Hence I firmly believe that incomputable mathematics is intrinsically less valuable than computable mathematics (allowing some exceptions for philosophical value and/or beauty), and that within computable mathematics, what can be computed in linear time is generally more valuable than what can be computed in polynomial (poly-) time, which in itself is more valuable than what can be computed in exponential (exp-) time, which in itself is more valuable than what can be computed just in theory. I've always been an exp-time mathematician. Almost everything I've worked on, finite-type invariants and invariants of certain 3-manifolds, categorification, matters related to associators and to free Lie algebras, etc., boils down to computable things, though they are computable in exp-time. My current project (joint with Roland van der Veen and continuing Lev Rozansky and Andrea Overbay) is poly-time, which puts it ahead of everything else I have done. IMHO it is also philosophically interesting and beautiful, but I'm biased. On to content: There is a standard construction that produces a knot invariant given a certain special element $R$ ("the $R$-matrix") in the second tensor power of some algebra $U$. Roughly speaking, one independent copy of $R$ is placed next to each crossing of a knot $K$, yielding an element in some high tensor power of $U$. Then the edges of $K$ provide ordering instructions for how to multiply together these tensor factors using the algebra structure of $U$ so as to get a $U$-valued knot invariant $Z$. Typically $U$ is the universal enveloping algebra of some semisimple Lie algebra $L$ (or some completed or quantized variant thereof). These algebras are infinite dimensional, and so $Z$ is not immediately computable. The standard resolution is to also choose a finite-dimensional representation $V$ of $L$ and to carry out all computations within $V$ and its tensor powers. This works incredibly well. In fact, almost all the ``knot polynomials'' that arose following the work of Jones and Witten, the Jones and coloured Jones polynomials, the HOMFLY-PT polynomial, the Kauffman polynomial, and more, arise in this way, and much if not all of ``quantum topology'' is derived from these seeds. Yet these polynomials take an exponential time to compute: within the computation high tensor powers of $V$ must be considered, the dimensions of such powers grow exponentially with the size of the input knot, and computations within exp-sized spaces take at least exp-time. The same criticism applies to almost everything else in quantum topology: whenever there is a ``braided monoidal category'' or a ``topological quantum field theory'' within some chain of reasoning, at some point high tensor powers of some vector spaces must be considered and the results become (at least) exp-time. (An alternative mean to extract computable information from $Z$ is to reduce modulo various ``powers of $h$'' filtrations on $U$. This yields the theory of finite type invariants. Individual finite type invariants are poly-time, but each single one is rather weak, and only when infinite sequences of finite type invariants are considered together, they become strong. Such sequences reproduce the aforementioned knot polynomials, but they are hard to compute). Our approach is different. We explain how one can ``fade out'' roughly a half of a given semisimple Lie algebra $L$ (namely its lower Borel subalgebra) by appropriately multiplying the structure constants that pertain to that half by some new coupling constant $b$. When $b=0$, the original $L$ collapses to a solvable Lie algebra inside which the computation of $Z$ is easy (as the name suggests, solvable algebras are easy to ``solve''). Alas at $b=0$ the result is always the same --- the classical (yet poly-time and very useful) Alexander polynomial. We find that in a formal neighborhood of $b=0$, namely in a ring in which $b^{k+1}=0$ for some natural number $k$, the invariant $Z$ remains poly-time to compute. By explicit experimentation with knots in the standard tables, the resulting poly-time invariants are very strong: with just $L=sl_2$ and $k=1$, the resulting invariant separates more knots than the exp-time HOMFLY-PT and Khovanov taken together. By both theory and experimentation, we know that our invariants give genus bounds for knots (hence they ``see'' some topology), and we have reasons to suspect that they may give a way to show that certain knots are not-ribbon, potentially assisting with the long-standing slice$=$ribbon conjecture. None of the above is written yet, though I have given many talks on the subject, and most are online with videos and handouts and running code. See \url{http://www.math.toronto.edu/drorbn/Talks/}. Within the time of the requested grant, I plan to complete my work on these poly-time invariants. Much remains to be done: writing from several perspectives, implementation for cases beyond $sl_2$ at $k=1$, a complete analysis of the relationship with genus and with the ribbon property, an analysis of the relationship with the Melvin-Morton-Rozansky expansion of the coloured Jones polynomial, and more. } \eject \par\noindent{\large\bf Poly-Time Knot Theory and Quantum Algebra \hfill Discovery Grant Proposal Summary} \vskip 2mm One of the major triumphs of mathematics in the 1980s, which lead 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, found a corner (figuratively speaking) of the vast state space of the quantum machinery used in knot theory, which can be described in just polynomial complexity, and which carries enough information to still speak to knot theory. The ``knot invariants'' constructed that way seem to be the strongest invariants we know that are computable even for very large knots. Our approach utilizes the fact that complicated symmetry groups often have much simpler ``contractions''. A well known example is the Lorentz group of relativity theory, which at small velocities contracts to the Galilean group of classical mechanics. In a similar manner we find that the symmetry algebras underlying the useful solutions of the Yang-Baxter equation, namely semi-simple algebras such as sl(n), have contractions that are ``solvable algebras'', and that the same operations that are exponentially complex for the original sl(n) symmetry become polynomially-complex (namely, much simpler) within and near these solvable contractions. Much remains to be done: implementation, documentation, application, generalization. I hope to achieve all that over this 5-year grant period. \eject \par\noindent{\Large\bf Poly-Time Knot Theory and Quantum Algebra \hfill Discovery Grant Proposal} \vskip 2mm %\begin{multicols}{2} Recently, Roland van der Veen and myself, following Lev Rozansky and Andrea Overbay~\cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}, presented a methodology~\cite{PPSA} for the construction of poly-time computable knot polynomials and constructed~\cite{PP1} the first poly-time computable knot polynomial since the Alexander polynomial of 1928~\cite{Alexander:TopologicalInvariants}. After a brief exposition of the methodology, I will explain why I am (very) excited, and what I hope to further achieve over this 5-year grant period. But first, {\bf Why is it exciting, even before the details?} Here and there math has immense philosophical value or beauty to justify the effort. Yet everyday math is mostly about, or should be about, ``doing useful things''. Deciding if $A$ has property $B$, counting how many $C$'s satisfy $D$, computing $E$. When $A$ and $B$ and $C$ and $D$ and $E$ are small, we do the computations on the back of an envelope and write them as ``Example 3.14'' in some paper. But these are merely the demos, and sooner or later we worry (or ought to worry) about bigger inputs. I'm more aware then most mathematicians (though perhaps less than many computer scientists), how much the complexity of obtaining the solution as a function of the size of the inputs matters. Hence I firmly believe that incomputable mathematics is intrinsically less valuable than computable mathematics (allowing some exceptions for philosophical value and/or beauty), and that within computable mathematics, what can be computed in linear time is generally more valuable than what can be computed in polynomial (poly-) time, which in itself is more valuable than what can be computed in exponential (exp-) time, which in itself is more valuable than what can be computed just in theory. With the exception of the Alexander polynomial, which has been thoroughly mined\footnote{Though newer and better still arises. For example, the techniques of~\cite{Bar-NatanSelmani:MetaMonoids} lead to the fastest known algorithm for the computation of the Alexander polynomial.}, and with the exception of the first few finite-type invariants~\cite{Bar-Natan:OnVassiliev, Bar-Natan:Polynomial}, which are rather weak, until recently~\cite{Bar-Natan:LesDiablerets-1608} all known knot invariants were harder-than-poly-time to compute\footnote{\label{foot:DivideAndConquer}Though divide-and-conquer methods reduce the computation time for the Jones and HOMFLY-PT polynomials~\cite{Jones:New, HOMFLY, PrzytyckiPrzytycki:PT} and possibly even for Khovanov homology~\cite{Bar-Natan:FastKh} to around $C\bbe^{c\sqrt{n}}$ where $n$ is the crossing number, and so these invariants can be computed for surprisingly large knots.}. So clearly, what we have in~\cite{PP1, PPSA} --- a rather strong poly-time-computable knot polynomial\footnote{How strong? As detailed in~\cite{Bar-Natan:Indiana-1611}, stronger than HOMFLY-PT and Khovanov taken together, at least for knots up to with 12 crossings.} and a methodology for further such --- is a priori exciting. Furthermore, our invariants extend to tangles, and are well-behaved under the basic tangle-theoretic operations of ``strand stitching'' and ``strand doubling'', and hence they carry topological information: a bound on the genus of a knot~\cite{Bar-Natan:Indiana-1611}, and what may be the best chance we have of showing that certain slice knots are not ribbon~\cite{K17}, hence resolving (negatively) the long-standing ``slice$=$ribbon'' conjecture~\cite{FoxMilnor:Singularities}. Finally, merely our suggestion (starting~\cite{Bar-Natan:Qinhuangdao-1507}) that some poly-time knot polynomials beyond the Alexander polynomial ought to exist is already generating both interest~\cite{Fiedler:1Cocycle} and competition~\cite{Przytycki:Vertigan} (both authors do not cite us explicitly, but were present in our talks~\cite{Bar-Natan:Toulouse-1705, Bar-Natan:GWU-1612} and were clearly influenced). {\bf Our methodology.} Since already the 1980s, there is a standard ``quantum algebra'' methodology that associates a framed knot invariant to certain triples $(U,R,C)$, where $U$ is a unital algebra and $R\in U\otimes U$ and $C\in U$ are invertible (see e.g.~\cite{Ohtsuki:QuantumInvariants}). For convenience, we recall this methodology an Aside. \begin{figure}\begin{center}\fbox{\begin{minipage}{0.99\linewidth}\sl \parpic[r]{\parbox[t]{1.9in}{\begin{center} \input{URCMethod.pdf_t} \newline $\displaystyle z(K) = \sum_{i,j,k} b_ia_jb_kCa_ib_ja_k$ \end{center}}} {\bf\sf Aside. The standard methodology on an example knot $K$.} Draw $K$ as a long knot in the plane so that at each crossing the two crossing strands are pointing up, and so that the two ends of $K$ are pointing up. Put a copy of $R=\sum a_i\otimes b_i$ on every positive crossing of $K$ with the ``$a$'' side on the over-strand and the ``$b$'' side on the under-strand, labeling these $a$'s and $b$'s with distinct indices $i,j,k,\ldots$ (similarly put copies of $R^{-1}=\sum a'_i\otimes b'_i$ on the negative crossings; these are absent in our example). Put a copy of $C^{\pm 1}$ on every cuap where the tangent to the knot is pointing to the right (meaning, a $C$ on every such cup and a $C^{-1}$ on every such cap). Form an expression $z(K)$ in $U$ by multiplying all the $a$, $b$, $C$ letters as they are seen when traveling along $K$ and then summing over all the indices, as shown. If $R$ and $C$ satisfy some conditions dictated by the standard Reidemeister moves of knot theory, the resulting $z(K)$ is a knot invariant. \picskip{1} Abstractly, $z(K)$ is obtained by tensoring together several copies of $R^{\pm 1}\in U^{\otimes 2}$ and $C^{\pm 1}\in U$ to get an intermediate result $z_0\in U^{\otimes S}$, where $S$ is a finite set with two elements for each crossing of $K$ and one element for each right-pointing cuap. We then multiply the different tensor factors in $z_0$ in an order dictated by $K$ to get an output in a single copy of $U$. \end{minipage} }\end{center}\end{figure} The best algebras with which to apply this methodology, at least as of 2017, are certain completions $\calU(\frakg)$ of the universal enveloping algebras of semi-simple Lie algebras $\frakg$ (or their quantizations). But these algebras are infinite dimensional, and the sum in the Aside is infinite and not immediately computable. The dogma solution is to pick a finite dimensional representation of $\frakg$ and use it to represent all the elements appearing in the Aside, effectively replacing the algebra by the algebra of endomorphisms of some finite dimensional vector space. This turns the sum finite; yet if the knot $K$ has $n$ crossings, our sum becomes a sum over $n$ indices $i_1,\ldots,i_n$. Thus there are exponentially-many summands to consider and it takes an exponential amount of time to compute $z(K)$.\textsuperscript{\ref{foot:DivideAndConquer},}\footnote{Note that almost any time the phrases ``braided monoidal category'' or ``TQFT'' are used within low dimensional topology, some tensor powers of some vector spaces need to be considered at some point, and dimensions grow exponentially. Thus our criticism applies in these cases too~\cite{Bar-Natan:Toulouse-1705}.} Alternatively, one may extract finite-type~\cite{Bar-Natan:OnVassiliev} information out of $z$ by reducing modulo appropriate filtrations of $U$ and its tensor powers. As already mentioned, the results are computable but weak. Our approach to the computation of $z(K)$ is different. Instead of working directly in $U^{\otimes S}$, we work in relatively small\footnote{Ranks grow polynomially in $|S|$.} spaces $\calF(S)$ of ``closed-form formulas for elements of $U^{\otimes S}$''. For this to work, we need to ensure that the fundamentals $R$ and $C$ would be described by ``closed-form formula'', and that the most basic operations necessary for the computation of $z$, namely multiplication of factors in $U^{\otimes S}$, can be implemented ``in closed form''. In practice, the kind of terms that appear within formulas for $R$ and $C$ are exponentials of the form $\bbe^{\xi x}$, where $x$ is a generator of $U$ and $\xi$ is a formal scalar variable, their iterated derivatives $(\partial_\xi)^k\bbe^{\xi x}=x^k\bbe^{\xi x}$, and exponentials of quadratics like $\bbe^{\lambda xy}$ or $\bbe^{\lambda x\otimes y}$, with scalar $\lambda$ and $x,y\in U$. We then need to multiply several such exponentials and differentiated exponentials, and we need to learn how to bring such products into some pre-chosen ``canonical order''. In the standard $U\sim\calU(\frakg)$ case, where $\frakg$ is semi-simple, this is complicated. Yet if $\frakg$ is solvable, this is often easy. Wouldn't it be nice if it was possible to approximate semi-simple Lie algebras with solvable ones? In this paper we exploit the little-known fact that this is (nearly) possible. Precisely, given a semisimple $\frakg$, there exists a Lie algebra $\frakg^\epsilon$ defined over the ring $\bbQ[\epsilon]$ of polynomials in a formal variable $\epsilon$ (in other words, $\frakg^\epsilon$ is a ``one-parameter family of Lie algebras''), so that \begin{enumerate} \item If $\epsilon$ is fixed to be some constant not equal to zero, then $\frakg^\epsilon$ is isomorphic to $\frakg^+\coloneqq\frakg\oplus\frakh$, which is the original $\frakg$ with an additional copy of its own (Abelian) Cartan subalgebra $\frakh$ added. \item At $\epsilon=0$, $\frakg^0$ is solvable. Furthermore, $\frakg^\epsilon$ is solvable in a formal neighborhood of $\epsilon=0$: for any natural number $k\geq 0$ the reduction $\frakg^{\leq k}$ of $\frakg^\epsilon$ to the ring $\bbQ[\epsilon]/(\epsilon^{k+1}=0)$ is solvable as a Lie algebra over $\bbQ$ (whose dimension is $(k+1)\dim\frakg$). \end{enumerate} As $k$ gets larger, the solvable $\frakg^{\leq k}$ is closer and closer to $\frakg^\epsilon$, as the reduction modulo $\epsilon^{k+1}=0$ means less and less, and so at least informally, $\frakg^{\leq k}\xrightarrow[k\to\infty]{}\frakg^+\sim\frakg$. It remains to sketch why $\frakg^\epsilon$ exists. Let $\frakg$ be a semisimple Lie algebra and let $\frakb^+$ and $\frakb^-$ be its upper and lower Borel subalgebras, respectively, Then $(\frakb^+)^\ast$ is $\frakb^-$, and as the latter has a Lie bracket, it follows that $\frakb^+$ has a co-bracket $\delta$. In fact, $\frakb^+$ along with its bracket $[\cdot,\cdot]$ and co-bracket $\delta$ is a ``Lie bialgebra'', and one may recover $\frakg^+=\frakg\oplus\frakh=\frakb^-\oplus\frakb^+$ as the ``Drinfel'd double'' $\calD(\frakb^+,[\cdot,\cdot],\delta)$ of $\frakb^+$ (see e.g.~\cite{EtingofSchiffman:QuantumGroups}). The axioms of a Lie bialgebra are homogeneous in $\delta$: meaning that $(\frakb^+,[\cdot,\cdot],\epsilon\delta)$ is again a Lie bialgebra for any scalar $\epsilon$, and one may set $\frakg^\epsilon\coloneqq\calD(\frakb^+,[\cdot,\cdot],\epsilon\delta)$, and the required properties are easy to check. Perhaps the most interesting is the solvability of $\frakg^0$: indeed $\frakg^0=I\frakb^+\coloneqq(\frakb^+)^\ast\rtimes\frakb^+$ with $(\frakb^+)^\ast$ regarded as an Abelian Lie algebra and $\frakb^+$ acts on $(\frakb^+)^\ast$ using the co-adjoint action, and then the solvability of $I\frakb^+$ easily follows from the solvability of $\frakb^+$. We studied the knot-theoretic significance of $\frakb^\ast\rtimes\frakb$ for a general Lie algebra $\frakb$ extensively in the context of ``w-knots'' in \cite{WKO1,WKO2,WKO3,WKO4,KBH}, and that these studies along with the observations in this paragraph were in some sense the starting points for our current study. {\red MORE.} %\end{multicols} \eject \par\noindent{\large\bf Poly-Time Knot Theory and Quantum Algebra \hfill Discovery Grant Proposal References} \vskip 2mm {\renewcommand{\section}[2]{}% \begin{thebibliography}{BN10} \bibitem[Al]{Alexander:TopologicalInvariants} J.~W.~Alexander, {\em Topological invariants of knots and link,} Trans.\ Amer.\ Math.\ Soc.\ {\bf 30} (1928) 275--306. \bibitem[BN1]{Bar-Natan:OnVassiliev} D.~Bar-Natan, \href{\myurl/papers/OnVassiliev/}{{\em On the Vassiliev Knot Invariants,}} Topology {\bf 34} (1995) 423--472. (Reported at the prestigious S\'eminaire Bourbaki. See P.~Vogel, {\em Invariants de Vassiliev des n\oe uds [d'apr\`es D.~Bar-Natan, M.~Kontsevich et V.~A.Vassiliev],} S\'eminaire Bourbaki {\bf 761} (1993) 1--17 \& Asterisque {\bf 216} (1993) 213--232). \bibitem[BN2]{Bar-Natan:Polynomial} D.~Bar-Natan, \href{\myurl/LOP.html#Polynomial}{{\em Polynomial Invariants are Polynomial,}} Mathematical Research Letters {\bf 2} (1995) 239--246. \bibitem[BN3]{Bar-Natan:FastKh} D.~Bar-Natan, \href{\myurl/papers/FastKh/}{{\em Fast Khovanov Homology Computations,}} Journal of Knot Theory and its Ramifications {\bf 16-3} (2007) 243--255. \bibitem[BN4]{KBH} D.~Bar-Natan, {\em \href{http://www.math.toronto.edu/~drorbn/papers/KBH/}{Balloons and Hoops and their Universal Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant,} } Acta Mathematica Vietnamica {\bf 40-2} (2015) 271--329, \arXiv{1308.1721}. \bibitem[BN5]{WKO4} D.~Bar-Natan, {\em Finite Type Invariants of W-Knotted Objects IV: Some Computations,} in preparation, \web{wko4}, \arXiv{1511.05624}. \bibitem[BN6]{Bar-Natan:Qinhuangdao-1507} D.~Bar-Natan, {\em Polynomial Time Knot Polynomials,} conference talks in Qinhuangdao and Aarhus, July 2015. Handouts and video at \web{q15} and \web{a15}. \bibitem[BN7]{Bar-Natan:LesDiablerets-1608} D.~Bar-Natan, {\em The Brute and the Hidden Paradise,} conference talk at Les Diablerets, August 2016. Handout and video at \web{ld16}. \bibitem[BN8]{Bar-Natan:Indiana-1611} D.~Bar-Natan, {\em A Poly-Time Knot Polynomial Via Solvable Approximation,} talk at Indiana University, November 2016. Handout and video at \web{ind}. \bibitem[BN9]{Bar-Natan:GWU-1612} D.~Bar-Natan, {\em On Elves and Invariants,} Conference talk at George Washington University, December 2016. Handout and video at \web{gwu}. \bibitem[BN10]{K17} D.~Bar-Natan, {\em Polynomial Time Knot Polynomial,} research proposal for the 2017 Killam Fellowship, \web{k17}. \bibitem[BN11]{Bar-Natan:Toulouse-1705} D.~Bar-Natan, {\em The Dogma is Wrong,} Conference talk in Toulouse, May 2017. Handout and video at \web{Toulouse}. \bibitem[BND1]{WKO1} D.~Bar-Natan and Z.~Dancso, \href {http://drorbn.net/AcademicPensieve/Projects/WKO1} {{\em Finite Type Invariants of W-Knotted Objects I: W-Knots and the Alexander Polynomial,}} Alg.\ and Geom.\ Top.\ {\bf 16-2} (2016) 1063--1133, \arXiv{1405.1956}. \bibitem[BND2]{WKO2} D.~Bar-Natan and Z.~Dancso, \href {http://drorbn.net/AcademicPensieve/Projects/WKO2} {{\em Finite Type Invariants of W-Knotted Objects II: Tangles and the Kashiwara-Vergne Problem,}} Math.\ Ann.\ {\bf 367} (2017) 1517--1586, \arXiv{1405.1955}. \bibitem[BND3]{WKO3} D.~Bar-Natan and Z.~Dancso, {\em Finite Type Invariants of W-Knotted Objects III: Double Tree Construction,} in preparation, \web{wko3}. \bibitem[BNS]{Bar-NatanSelmani:MetaMonoids} D.~Bar-Natan and S.~Selmani, {\em Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial,} Journal of Knot Theory and Its Ramifications {\bf 22-10} (2013), \web{BNS}. \bibitem[BV1]{PP1} D.~Bar-Natan and R.~van~der~Veen, {\em A Polynomial Time Knot Polynomial,} \arXiv{1708.04853}. \bibitem[BV2]{PPSA} D.~Bar-Natan and R.~van~der~Veen, {\em Poly-Time Knot Polynomials Via Solvable Approximations,} in preparation, \web{BV2}. \bibitem[ES]{EtingofSchiffman:QuantumGroups} P.~Etingof and O.~Schiffman, {\em Lectures on Quantum Groups,} International Press, Boston, 1998. \bibitem[Fi]{Fiedler:1Cocycle} T.~Fiedler, {\em Knot Polynomials from 1-Cocycles,} \arXiv{1709.10332}. \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[HOMFLY]{HOMFLY} J.~Hoste, A.~Ocneanu, K.~Millett, P.~Freyd, W.~B.~R.~Lickorish, and D.~Yetter, {\em A new polynomial invariant of knots and links,} Bull.\ Amer.\ Math.\ Soc.\ {\bf 12} (1985) 239--246. \bibitem[Jo]{Jones:New} V.~F.~R. Jones, {\em A polynomial invariant for knots via von Neumann algebras,} Bull.\ Amer.\ Math.\ Soc.\ {\bf 12} (1985) 103--111. \bibitem[Oh]{Ohtsuki:QuantumInvariants} T.~Ohtsuki, {\em Quantum Invariants,} Series of Knots and Everything {\bf 29}, World Scientific 2002. \bibitem[Ov]{Overbay:Thesis} A.~Overbay, {\em Perturbative Expansion of the Colored Jones Polynomial,} University of North Carolina PhD thesis, \web{Ov}. \bibitem[Pr]{Przytycki:Vertigan} J.~H.~Przytycki, {\em The First Coefficient of Homflypt and Kauffman Polynomials: Vertigan Proof of Polynomial Complexity using Dynamic Programming,} \arXiv{1707.07733}. \bibitem[PT]{PrzytyckiPrzytycki:PT} J.~H.~Przytycki and P.~Traczyk, {\em Conway Algebras and Skein Equivalence of Links,} Proc.\ Amer.\ Math.\ Soc.\ {\bf 100} (1987) 744--748. \bibitem[Ro1]{Ro} L.~Rozansky, {\em A contribution of the trivial flat connection to the Jones polynomial and Witten's invariant of 3d manifolds, I,} Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}. \bibitem[Ro2]{Rozansky:Burau} L.~Rozansky, {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton Expansion of the Colored Jones Polynomial,} Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}. \bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky, {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,} \arXiv{math/0201139}. \end{thebibliography}} \eject \par\noindent{\large\bf Poly-Time Knot Theory and Quantum Algebra \hfill Discovery Grant Proposal To Do List} \vskip 2mm \begin{itemize} \item Clear this list and remove this page. \item Proposal is up to 5 pages, list of references up to 2. \item Clean the hyperlinks in the bibliography. \item Is there some big-name-quote along the lines of ``computable better than not''? \item HQP Training Plan (5000 characters, TXT). \item Most Significant Contributions (7500 characters, TXT). \item Additional Information on Contributions (2500 characters, TXT). \item Budget justification (2 pages, PDF). \item 4 samples of research contributions (PDF). \item Attach CCV. \item Renumber pages and partition. \item Reread instructions pages. \end{itemize} \end{document}