Genus of one dimensional curves

In summary, The conversation discusses the concept of genus in physics and differential geometry. The term "genus one potentials" is used to describe cubic and quartic anharmonic oscillators, while "higher genus potentials" refers to higher order anharmonic oscillators. The concept of genus is also related to Riemann surfaces and can be calculated using the genus-degree formula. A helpful explanation for computing genus can be found at the provided link.
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detre
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Hello,

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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Likes Ben Niehoff

What is a genus of one dimensional curve?

A genus of one dimensional curve is a mathematical concept that describes the number of holes or handles in a curve. It is a topological property that is used to classify and distinguish different types of curves.

How is the genus of a one dimensional curve calculated?

The genus of a one dimensional curve can be calculated using the Euler characteristic formula, which is given by g = 1 - (n/2), where g is the genus and n is the number of boundary components of the curve. Alternatively, it can also be calculated using the Riemann-Hurwitz formula for compact Riemann surfaces.

What is the significance of the genus in geometry?

The genus of a one dimensional curve is an important topological invariant in geometry. It helps to classify and distinguish different types of curves, such as circles, ellipses, and hyperbolas. It also has applications in other fields of mathematics, such as algebraic geometry and differential geometry.

Can the genus of a one dimensional curve be negative?

Yes, the genus of a one dimensional curve can be negative. This occurs when the curve has a larger number of boundary components than expected based on its Euler characteristic. In such cases, the curve is said to have a negative genus.

How does the genus of a one dimensional curve relate to its complexity?

In general, the higher the genus of a one dimensional curve, the more complex it is. This is because a higher genus indicates a larger number of holes or handles in the curve, which makes it more intricate and difficult to study. However, this is not always the case as some curves with low genus can also have complex shapes and properties.

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