The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it.
A torus has genus one, as does the surface of a coffee cup.
The genus of a knotK is the least integer g = g(K) such that K is the boundary of a Seifert surface of genus g.
An unknot (also called a trivial knot) O–which is, by definition, the boundary of a disc embedded in the 3-sphere S3–has genus zero, and any knot of genus zero is an unknot.
The trefoil knot has genus one, as does the figure-eight knot.
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.
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. How about this in matrix theory? could anyone say....
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