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The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it. For instance: A ball has genus 0. A solid torus D 2 × S 1 has genus 1.
The term triple torus is also occasionally used to denote a genus 3 surface. [7] [5] The Klein quartic is a compact Riemann surface of genus 3 with the highest possible order automorphism group for compact Riemann surfaces of genus 3. It has 168 orientation-preserving automorphisms, and 336 automorphisms altogether. Several genus 3 surfaces
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
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Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ) (,) = (+ ) (,) = using angular coordinates θ, φ ∈ [0, 2π), representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius R is the distance from the center of the tube to ...
Two different pants decompositions for the surface of genus 2. The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface of genus with boundary components to be () = +, and for a non-connected surface take the sum over all components.
A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
A radially symmetric 7-colored torus – regions of the same colour wrap around along dotted lines An 8-coloured double torus (genus-two surface) – opposite circles are linked by handles A 6-colored Klein bottle, the only exception to the Heawood conjecture
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