What is the genus of an object?
A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface.
What is a non orientable shape?
A space is non-orientable if “clockwise” is changed into “counterclockwise” after running through some loops in it, and coming back to the starting point. This means that a Geometric shape, such as , that moves continuously along such a loop is changed in its own mirror image. .
Is a torus orientable?
Orientable surfaces are surfaces for which we can define ‘clockwise’ consistently: thus, the cylinder, sphere and torus are orientable surfaces. In fact, any two-sided surface in space is orientable: thus the disc, cylinder, sphere and n-fold torus, all with or without holes, are orientable surfaces.
Is a double torus orientable?
A double torus is a topological surface with two holes, formed from the connected sum of two tori. It has an orientable genus of 2 and an Euler characteristic of -2.
What is the genus of graph?
The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
What is the difference between species and genus?
A genus consists of a large number of organisms, whereas species consists of a fewer number of organisms. The best example is animals like zebra, horses, and donkeys which belong to the same Genus “Equss”. Meaning all the different species of zebra, donkey, and horses all belong to Equss.
Can a Möbius strip exist?
A Möbius strip can be created by taking a strip of paper, giving it an odd number of half-twists, then taping the ends back together to form a loop. The Möbius strip has more than just one surprising property. For instance, try taking a pair of scissors and cutting the strip in half along the line you just drew.
How many dimensions is a Möbius strip?
two-dimensional
The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface that is not orientable. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability.
Can a Klein bottle exist?
A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole. In this sense, a Klein Bottle is a 2-dimensional manifold which can only exist in 4-dimensions! Alas, our universe has only 3 spatial dimensions, so even Acme’s dedicated engineers can’t make a true Klein Bottle.
Is a Plane a closed surface?
The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces.
How many handles does a double torus have?
two handles
Since this is like adding a handle we will sometimes refer to the torus as the sphere with one handle, the double torus as the sphere with two handles, etc. Figure II. 2: Left: the projective plane, P2, obtained by gluing a disk to a Möbius strip.
Is the double torus a surface?
The term double torus is occasionally used to denote a genus 2 surface. A non-orientable surface of genus two is the Klein bottle. The Bolza surface is the most symmetric Riemann surface of genus 2, in the sense that it has the largest possible conformal automorphism group.
Which is the non orientable genus of a graph?
Graph theory. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n ). (This number is also called the demigenus .) The Euler genus is the minimal integer n such…
How is the genus of a closed surface defined?
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. A real projective plane has non-orientable genus one. A Klein bottle has non-orientable genus two. The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K.
How is the genus of a handlebody defined?
The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary. 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.
What is the meaning of genus in mathematics?
Genus (mathematics) In mathematics, genus (plural genera) has a few different, but closely related, meanings. The most common concept, the genus of an (orientable) surface, is the number of “holes” it has, so that a sphere has genus 0 and a torus has genus 1.
