What's the difference in meaning and usage between Twin and twin?

So s is in the back,

It's twins',

It means: twins, here is the singular, one of the twins.

Twins are a complex number,

So s is in front,

Are twins,

It means: twins, here is the plural.

For example, Teacher's Day.

This is not just a teacher's holiday, but a holiday for all teachers, so add the plural s in front of it and type'.