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  \lhead{\footnotesize\href{\myurl}{Dror Bar-Natan}: \href{\pensieve}{Academic Pensieve}: \href{\pensieve/Projects}{Projects}: \href{http://drorbn.net/K17}{Killam-2017}:}
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\par\noindent{\Large\bf Dror Bar-Natan \hfill Most Significant Contributions and Publications}
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%{\tiny  Identify a maximum of five (5) contributions, with a maximum total length of one page for all five, that best highlight your contribution to research or your research-related activities, defining the impact and relevance of each. A contribution is understood to be a publication, literary or artistic work, conference, patent or intellectual property right, contract or creative activity, commission, etc. Your complete description may relate to: the organization; the position or activity type and their description; start and end dates; and the basis on which this contribution is significant (i.e. relevance, target community and impact). One page maximum.
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\begin{quote} \fbox{\em
  Construct, implement, and document poly-time computable polynomial invariants of knots and tangles
}\end{quote}

\noindent{\bf\large Most Significant Contributions.}\hfill(in relation to the proposed project)

\vskip 2mm\noindent{\bf Contribution A.} My paper {\em On the Vassiliev Knot Invariants}~\cite{Bar-Natan:OnVassiliev}. This is an old paper, yet it is worth noting as in it I have first laid the foundations for the substitution ``diagram-valued invariants of knots'' instead of ``an invariant for each Lie algebra and representation thereof''. This substitution is one of the keys to the realization of the goals of my project: the Lie-theory approach is intrinsically exponential time, yet as I argue in the ``Detailed Project Description'', the diagrammatic approach (properly modernized) affords quotients that are of polynomial complexity.

\vskip 2mm\noindent{\bf Contribution B.} My paper with S.~Garoufalidis {\em On the Melvin-Morton-Rozansky Conjecture}~\cite{Bar-NatanGaroufalidis:MMR}. This is the first place where the Alexander polynomial (poly-time) was proven to be dominated by the coloured Jones polynomial (highly exponential). One aspect of my proposed project is the realization that there ought to be further poly-time sections of the coloured Jones polynomial.

\vskip 2mm\noindent{\bf Contribution C.} My paper {\em Fast Khovanov Homology Computations}~\cite{Bar-Natan:FastKh} which is a direct continuation of~\cite{Bar-Natan:Cobordism}. This paper describes my mathematically-sophisticated methodology for the computation of Khovanov homology. While not poly-time, it is many orders of magnitude more efficient than the naive approach, and it made Khovanov homology computable even for rather large knots (knots with up to 50-70 crossings). Regrettably, Khovanov homology fails criteria C5 of my ``Detailed Project Description'', and hence the need to do even better.

\vskip 2mm\noindent{\bf Contribution D.} My paper with Z.~Dancso {\em Finite Type Invariants of W-Knotted Objects I}~\cite{WKO1} and its sequels in the same series {\em II}~\cite{WKO2}, {\em III} (in preparation), and {\em IV} (\arXiv{1511.05624}). These papers fully analyze the quotient $\calA^w$ that is mentioned in the ``Detailed Project Description'', relating it to 4-dimensional topology, and to Lie algebras and the Kashiwara-Vergne problem. Most importantly from the perspective of the current proposal is that in this series we set up an intricate theoretical framework for computations in $\calA^w$. The purpose of the current project is to lift that framework one level higher, to the quotient denoted $\calA^{2,2}$ in the detailed description.

\vskip 2mm\noindent{\bf Contribution E.} I believe in complete transparency and open access, and therefore nearly every talk that I have given in the last 15 years, many with direct relations to this project, was accompanied by an openly-available informational handout and many of the talks are available on video. See \web{Talks}. Likewise almost every internal note or computer program that I have written in relation to this project or otherwise in the last 8 years is at \web{AP}. Altogether my personal web site \web{me} contains several thousand documents and serves as a resource and repository for the knot theory community.

\vskip 3mm\noindent{\bf\large Publications.}\hfill(In mathematics authors are normally listed alphabetically)

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