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\begin{document} %\latintext
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\title{Over then Under Tangles}

\author{Dror~Bar-Natan}
\address{
  Department of Mathematics\\
  University of Toronto\\
  Toronto Ontario M5S 2E4\\
  Canada
}
\email{drorbn@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/drorbn}

\author{Zsuzsanna Dancso}
\address{
  School of Mathematics and Statistics\\
  The University of Sydney\\
  Eastern Ave\\
  Camperdown NSW 2006, Australia
}
\email{zsuzsanna.dancso@sydney.edu.au}
\urladdr{http://zsuzsannadancso.net}

\author{Roland~van~der~Veen}
\address{
  University of Groningen, Bernoulli Institute\\
  P.O. Box 407\\
  9700 AK Groningen\\
  The Netherlands
}
\email{roland.mathematics@gmail.com}
\urladdr{http://www.rolandvdv.nl/}

\date{First edition Jul.\ 16, 2020, this edition \today. Other/online versions:~\cite{Self}.}

%\subjclass[2010]{57M25}
\subjclass[2020]{57K12, 20F36, 20F38}
\keywords{knots, braids, virtual braids, tangles, virtual tangles,
diamond lemma, extraction graphs, Drinfel'd double}

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\draftcut \section{Acknowledgement} We wish to thank P.~Bellingeri,
A.~Referees, D.~Thurston, and B.~Wiest for comments and suggestions,
and especially O.~Chterental for spotting a major gap in an earlier
version of this paper. This work was partially supported by NSERC grant
RGPIN-2018-04350, by the University of Sydney Visiting Scholar Scheme, and
by the Chu Family Foundation (NYC).

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