Genus of one dimensional curves

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In a physics paper, I have encountered an expression about genus of one dimensional anharmonic oscillators. More specifically, they classify cubic and quartic anharmonic oscillator as "genus one potentials" and higher order anharmonic oscillators as "higher genus potentials".

I am new in differential geometry and topology but I know basic notion of genus in Riemann surfaces. My question is how is a genus defined for a one dimensional curve and how should I count them?

Thanks in advanced!

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