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I am learning differential geometry by programming them and seeing their shapes. But topology is absolutely mysterious to me. For example, a sphere $$ x^2+y^2+z^2=r^2 $$ has genus 0 (no holes).

and a torus $$ \left(R- \sqrt{x^2+y^2} \right)^2+z^2=r^2 $$ has genus 1 (with one hole).

But is there a formula than can actually derive the number 0, and 1 from the above equations? In other words, how to compute the genus of an algebraic surface?

p.s. Most functions I know return a real number (e.g. $\sin$, $\cos$, $\exp$, etc), so I am very curious about how a function transform a surface representation into an integer. Or, Does "a surface has genus 1.5 " make any sense?

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    See [this question](http://math.stackexchange.com/questions/278987/an-analogue-of-degree-genus-formula-for-surfaces). – 2017-02-17
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    how to determine the value of $d$ and $n$, from a surface representation such as $x^2+y^2+z^z=r^2$? – 2017-02-17

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Any compact Riemann surface $R$ is homeomorphic to a sphere with handles. The number $g$ of handles is called the genus of $R$. With this standard definition we see that the first example, the sphere without handles, has genus zero, whereas the torus can be deformed (the hole becomes a handle) to a sphere with $1$ handle. Hence its genus is equal to $1$. There are also several other methods to compute the genus $g$ of a compact Riemannian manifold, e.g., $g=(\chi(R)-2)/2$, where $\chi(R)$ is the Euler characteristic. See also for "Riemann-Hurwitz formula", or "Riemann-Roch".

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It isn't possible to provide a general formula that computes genus of surface from its coordinate equations since genus is the topological (not just geometric) invariant and parametrization may vary under homeomorphisms.