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The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation.
The genus-degree formula for plane curves can be deduced from the adjunction formula. [2] Let C ⊂ P 2 be a smooth plane curve of degree d and genus g. Let H be the class of a hyperplane in P 2, that is, the class of a line. The canonical class of P 2 is −3H.
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...
Colloquially speaking, the genus of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first Betti number , i.e., half of the C {\displaystyle \mathbb {C} } -dimension of the first singular homology group H 1 ( X , C ...
A g-torus knot is a closed curve drawn on a g-torus. More technically, it is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus g handlebody in S³ (whose complement is also a genus g handlebody). If a link is a subset of a genus two handlebody, it is a double torus link. [14]
Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued together i.e. a torus. So the genus of an elliptic curve is 1. The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations.
The most common examples are the curves X(N), X 0 (N), and X 1 (N) associated with the subgroups Γ(N), Γ 0 (N), and Γ 1 (N). The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron. The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann ...