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\parbox[b]{3.25in}{
  {\LARGE\bf Cheat Sheet Plane Curves}
  }
\hfill\parbox[b]{4.75in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated Jan 2, 2014; modified \today
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{\bf \foreignlanguage{russian}{ПЕРЕСТРОЙКА}} (perestroika): reconstruction, change-over, improvement, realignment, reformation.

{\bf Comment.} ``Virtual plane curves'' make sense; I don't know their G-FT theory and its relationship with the u case.

{\bf \cite{CD}:} $J^+-J^-=n=\#\text{double points}$, and
\[ J^-(\Gamma)=1-\sum_{\tilde{C}}\ind^2_{\tilde\Gamma}(\tilde{C})\chi(\tilde{C}), \]
where $\tilde{C}$ runs over regions in the oriented smoothing $\tilde{\Gamma}$ of $\Gamma$ and $\chi$ denotes Euler characteristic.

Arnold's normalization. $St(K_0)=0$, $J^+(K_0)=0$, $J^-(K_0)=-1$; for $i>0$, $St(K_i)=i-1$, $J^+(K_i)=-2(i-1)$, $J^-(K_i)=-3(i-1)$; from \cite{Ar}:
\[ \includegraphics[width=\columnwidth]{ArnoldsNormalization.png} \]

{\small
\vskip 2mm{\bf References.}
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\bibitem[Ar]{Arnold:PlaneCurves} V.I.~Arnold,
  {\em Topological Invariants of Plane Curves and Caustics,}
  American Mathematical Society, 1994.

\bibitem[Ca]{Carter:Classifying} Carter: {\em Classifying Immersed Curves.}

\bibitem[CD]{CD} Chmutov, Duzhin: {\em Explicit Formulas for Arnold's Generic Curve Invariants.}

\bibitem[FT]{FennTaylor:IntroducingDoodles} R.~Fenn and P.~Taylor,
  {\em Introducing Doodles,}
  in {\em Topology of Low-Dimensional Manifolds, Proceedings of the Second Sussex Conference, 1977}, Springer 1979.

\bibitem[Go1]{Goryunov:FiniteOrderAndJ} Goryunov: {\em Finite Order Invariants of Framed Knots in a Solid Torus and in Arnold's $J^+$-Theory of Plane Curves.}

\bibitem[Go2]{Goryunov:WaveFronts} Goryunov: {\em Plane Curves, Wave Fronts and Legendrian Knots.}

\bibitem[Go3]{Goryunov:VassilievAndArnold} Goryunov: {\em Vassiliev Type Invariants in Arnold's $J^+$-Theory of Plane Curves without Direct Self-Tangencies.}

\bibitem[Gu]{Gulde:Classification} Gulde: {\em Classification of Plane Curves.}

\bibitem[Kh1]{Khovanov:DoodleGroups} M.~Khovanov,
  {\em Doodle Groups,}
  Trans.\ Amer.\ Math.\ Soc.\ {\bf 349-6} (1997) 2297--2315.

\bibitem[Kh2]{Khovanov:Tabachnikov} Khovanov: {\em Some Remarks on Tabachnikov's Invariants of Plane Curves.}

\bibitem[Me1]{Merkov:Ornaments} Merkov: {\em Finite-Order Invariants of Ornaments.}

\bibitem[Me2]{Merkov:FlatBraids} Merkov: {\em Vassiliev Invariants Classify Flat Braids.}

\bibitem[Me3]{Merkov:SegementArrowDiagrams} A.B.~Merkov,
  {\em Segment-Arrow Diagrams and Invariants of Ornaments,}
  Sbornik: Mathematics {\bf 191-11} (2000) 1635--1666.

\bibitem[Me4]{Merkov:PlaneCurvesAndDoodles} A.B.~Merkov,
  {\em Vassiliev Invariants Classify Plane Curves and Doodles,}
  Sbornik: Mathematics {\bf 194-9} (2003) 1301-1330.

\bibitem[Ng]{Ng:ContactGeometry} L.~Ng: {\em Plane Curves and Contact Geometry.}

\bibitem[No1]{Nowik:PlaneCurves} T.~Nowik: {\em Order One Invariants of Plane Curves.}

\bibitem[No2]{Nowik:SphericalCurves} T.~Nowik: {\em Order One Invariants of Spherical Curves.}

\bibitem[Oc]{Ochiai:PlanneCurves} Ochiai: {\em Invariants of Plane Curves and Polyak-Viro Type Formulas for Vassiliev Invariants.}

\bibitem[Oz]{Ozawa:PlaneCurves} Ozawa: {\em Finite Order Topological Invariants of Plane Curves.}

\bibitem[Po]{Polyak:CurvesAndFronts} Polyak: {\em Invariants of Curves and Fronts Via Gauss Diagrams.}

\bibitem[Se]{Selwat:Semilocal} Selwat: {\em The First Order Semilocal Vassiliev Invariants of Plane Curves.}

\bibitem[Sh]{Shumakovich:Explicit} Shumakovich: {\em Explicit Formulas for Strangeness} (Russian).

\bibitem[Ta]{Tabachnikov:TriplePointFree} S.~Tabachnikov,
  {\em Invariants of Smooth Triple Point Free Plane Curves,}
  Jour.\ of Knot Theory and its Ramifications {\bf 5-4} (1996) 531--552.

\bibitem[Va1]{Vassiliev:TriplePointFree} V.A.~Vassiliev,
  {\em On Finite Order Invariants of Triple Point Free Plane Curves,}
  1999 preprint, \arXiv{1407.7227}.

\bibitem[Va2]{Vassiliev:Ornaments} V.A.~Vassiliev,
  {\em Invariants of Ornaments,}
  Adv.\ in Soviet Math.\ {\bf 21} (1994) 225--262.

\bibitem[Vi1]{Viro:GenericImmersions} Viro: {\em Generic Immersions of Circle to Surfaces and Complex Topology of Real Algebraic Curves.}

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