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\title{A Perturbed-Alexander Invariant}

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

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

\date{First edition June 24, 2022. This edition \today.}

\dedicatory{Dedicated to the memory of V.~F.~R.~Jones, 1952--2020, a friend and a mentor.}

\makeatletter
\@namedef{subjclassname@2020}{\textup{2020} Mathematics Subject Classification}
\makeatother
\subjclass[2020]{Primary 57K14, secondary 16T99}
\keywords{
  Alexander polynomial,
  Jones polynomial,
  knot invariants,
  loop expansion,
  poly-time computations,
  quantum algebra,
  Reidemeister moves,
  ribbon knots,
  Seifert surfaces,
  solvable approximation%
}

\thanks{This work was partially supported by NSERC grant RGPIN-2018-04350
and by the Chu Family Foundation (NYC). It is available in electronic
form, along with source files and a verification {\sl Mathematica}
notebook at \url{http://drorbn.net/APAI} and at \arXiv{2206.12298}.}

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