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n-HOLED TORUS



The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set .
Any orientable connected compact surface without boundary is homeomorphic to an n-torus.
The Euler characteristic of the n-torus is equal to 22n.
 
 
 
The double torus is informally called "pretzel". On the right, a mathematical pretzel composed of two loops that are Viviani curves and an arc of a circle. 

 
An algebraic pretzel of degree 4, with 2 planes of symmetry, and equation ; this equation was knocked up from that of the figure-eight, in order to thicken it.
Starting from the lemniscate of Bernoulli, we get the figure below, with equation:
.
A triple algebraic torus, based on the regular trifolium, with equation: .
A quadruple algebraic torus, with equation: .

 
The n-holed torus is also informally called "fougasse": opposite, a fougasse with 6 holes.

Do not mistake the n-torus  with the n-dimensional torus,; in particular, .


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© Robert FERRÉOL  2017