Talk:Genus (mathematics)

Open Annulus has genus 0

If you can include boundary points in the closed curve, you can cut the Annulus...

Using an annulus as an example of a genus zero is confusing to me. Perhaps an encyclopedia should not be confusing, and annulus should be removed from the list of examples.

If the annulus as genus zero is important to grasping the concept, then it could be added at the end of the article for clarification. In the interim, please don't add to my confusion. —Preceding unsigned comment added by Softcafe (talk • contribs) 02:41, 2 October 2009 (UTC)

A pretzel has genus 2, and so has the number 8 and the letter B.

Shouldn't a pretzel be genus 3? Most pretzels I've seen have two big semicircular holes and one smaller triangular one.

pretzel!!! MotherFunctor 04:32, 15 May 2006 (UTC)

The one in Burg has genus 2. (But the one seen as a bakery emblem in Ribe has genus 4.) Alfa-ketosav (talk) 10:45, 7 May 2020 (UTC)

It is all wrong

The page is all wrong, I always thoght that genus is an invariant for surfaces only and nuber of holes is not metter, S^2, D^2, and cilinder all have geus zero. I thought to change it, but it seems that there is no standard agreement on when genus is defined, for sure oriented surfeces are included, it is used sometimes for nonorented, but with not oreinted it seems there is no standard def... Look at [1] and [2]

Tosha 23:49, 28 May 2004 (UTC)

I agree, see below. MotherFunctor 04:31, 15 May 2006 (UTC)

genus of the Bottle of Klein

The Bottle of Klein is a non-orientable surface of genus 1, not 2 as stated on this page. It is correct on [3].

(Citing a wiki page as reference is very weak, as is no signature!) Mathworld gives genus as the maximum number of non-intersecting Jordan curves such that their complement in the surface is path connected. This substantiates the above section "it's all wrong" and substantiates Klein's bottle is genus 2. MotherFunctor 04:30, 15 May 2006 (UTC)

Write K for the Klein bottle and P for the real projective plane. It can easily be shown that K = P # P, thus the (non-orientable) genus of K is 2. Morana (talk) 21:23, 7 April 2008 (UTC)

Genus: mathematics or geometry

I'ld quite like to move this page to something like Genus (geometry), and turn it into a mathdab instead. There's another perfectly good use of the word genus in mathematics, namely in the theory of quadratics forms and by extension in algebraic number theory. There is also the concept of Genus of a multiplicative sequence which is referred to here. Richard Pinch (talk) 10:43, 29 June 2008 (UTC)

Separating out Geometry from Algebra would very helpful for readers like me who come looking for algebra and then take some time to realise that the definitions are disjoint. Dan Shearer (talk) 08:07, 5 October 2022 (UTC)

Definition of cuttings

• The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected.
  • How is cuttings defined? If one cuts a small circle out of the surface of a torus, one gets two disconnected pieces, although the torus is of genus 1. Thus, which part of the definition do I misinterpret? Thanks, --Abdull (talk) 08:27, 28 July 2008 (UTC)
    • No. If you make one cut you end up with a cylinder, which is connected. Only with two cuts do you end up with two pieces. Turiacus (talk) 10:28, 28 July 2008 (UTC)

If you make one cut you may end up with a cylinder, or you may have two pieces: a torus with a disk cut out, and the disk. The fact that you can cut once without disconnecting the surface is what makes the genus. Zaslav (talk) 05:47, 29 July 2009 (UTC)

negative genus?

So I've never studied topology, but is it possible to have an object with a negative genus? If not, shouldn't the opening sentence of the "topology" section read "The genus of a connected, orientable surface is a positive integer representing..." ? Origamidesigner (talk) 19:08, 26 November 2014 (UTC)

non-negative integer at best, since genus 0 is possible. Apart from that, it could be changed, but I don't the the improvement (nor, of course, the worsening) of the article in doing so. --Ulkomaalainen (talk) 20:58, 17 March 2015 (UTC)

Negative genus is indeed possible. An interior void in a solid of genus zero subtracts one from the genus. Adding a through hole (one which passes through the exterior surface to the interior surface of the void adds one to the genus and results in an object of genus 0, therefore it must prior have had negative one genus. — Preceding unsigned comment added by 136.1.1.102 (talk) 18:53, 21 April 2015 (UTC)

Redundant

At the examples of the genus-$n$ surfaces, there is the same description as somewhere above. Alfa-ketosav (talk) 16:40, 9 August 2019 (UTC)

Error under "Algebraic geometry"

The phrase When X is an algebraic curve with field of definition the complex numbers is missing some words, which I am sure a mathematician can easily fix. Thanks.

Dan Shearer (talk) 08:03, 5 October 2022 (UTC)