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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.
The curve complex of a surface is a complex whose vertices are isotopy classes of simple closed curves on . The action of the mapping class groups Mod ( S ) {\displaystyle \operatorname {Mod} (S)} on the vertices carries over to the full complex.
The moduli space,, (,) is not compact, because a sequence of curves can degenerate to a singular curve, which is not in the moduli space as defined above. This happens, for example, when the energy of f {\displaystyle f} (meaning the L 2 norm of the derivative) concentrates at some point on the domain.
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.
The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be ...
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S.The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups.
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]
The genus (sometimes called the 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 − g , where g is the non-orientable ...