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  {\LARGE\bf Cheat Sheet Plane Curves}
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\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
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\bibitem[0]{0} Khovanov, Merkov, Ng, Nowik, Ochiai, Ozawa, Polyak, Selwat, Shumakovich, Tabachnikov, Vassiliev, Viro.

\bibitem[Ar]{Ar} Arnold: {\em Topological Invariants of Plane Curves and Caustics.}

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

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

\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.}

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