Understanding Disks-and-Bands Surfaces: Genus and Boundary Components

In summary, the conversation discusses the representation of orientable surfaces in terms of disks and bands. The upper part of the surface contributes one boundary component as a whole, while each pair of handles in the lower part increases the genus by 2. Attaching a single handle in the lower part increases both the number of boundary components and the overall genus by 1. The accuracy of this information is being verified and further updates may be provided.
  • #1
WWGD
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Hi all, I was reviewing some old material on the representation of orientable surfaces
in terms of disks and bands , in page 2 of:


http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf

Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into
an upper part and a lower part. Consider pairs in the upper part ; each represents a 2-torus with a disk removed. Consider too, individual handles in the lower part; each attached handle is basically a disk D^2 with a smaller disk d^2 removed. Then:

1) The "upper part" contributes one boundary component as a whole ; every pair of handles contributes 2 to the genus, i.e., every time we attach a pair of handles as in the upper part, the genus increases by 2 , since we can add a pair of essential , i.e., non-separating loops l_1, l_2 , each going exactly once around each of the respective handles (and, of course, any other loop about these handles would be homologous to either l_1 or to l_2).

2) Attaching a single handle as in the lower part increases both the number of boundary components by 1 ( the inner loop within the handle, representing a small disk d^2 removed from
D^2 ), and also increases the overall genus by 1, since we now have a new essential loop for each handle, this loop is the one that winds around the inner hole.

Is this correct?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
Thanks, Greg, I'm trying to freshen up on the topic see if I can ask a more precise question. I'll keep you posted.
 

What is the definition of "genus" in relation to disk-and-bands surfaces?

The genus of a disk-and-bands surface refers to the number of holes or handles that can be created on the surface without breaking it. It is a topological invariant, meaning it does not change even if the surface is stretched or deformed.

How do you determine the genus of a disk-and-bands surface?

To determine the genus of a disk-and-bands surface, you can use the Euler characteristic formula, which states that the number of vertices minus the number of edges plus the number of faces equals 2 minus twice the genus. Alternatively, you can also count the number of handles or holes on the surface.

What are boundary components in disk-and-bands surfaces?

Boundary components are the edges or boundaries of a disk-and-bands surface that are not connected to any other edges. They are often represented as dashed lines in diagrams and can affect the genus of the surface.

Can a disk-and-bands surface have more than one genus?

Yes, a disk-and-bands surface can have multiple genus if it has multiple holes or handles. The genus of each hole or handle is added together to determine the overall genus of the surface.

How are disk-and-bands surfaces used in real-world applications?

Disk-and-bands surfaces have many applications in mathematics, computer graphics, and engineering. They are used to model complex shapes and structures, such as knots, meshes, and surfaces with holes. They are also useful in studying the behavior of materials and designing objects with specific properties.

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