Topics from today's class:
- Partitions of surfaces
- Euler characteristic
- Euler characteristic for connected sums
- 4.5, 4.6, 4.7, 4.8, 4.9, 4.10
Read for next class:
Write one sentence about the reading assignment. You cannot repeat what another student has posted.
I learned that surfaces can be uniquely identified by Euler Characteristic, number of boundary components, and orientability.
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I think it would be interesting to look at more surfaces like the Mobius band that arent orientable.
How exactly do the surfaces with boundary component of a knot help us? Do distinct knots map to distinct surfaces (up to homeomorphism)? Does every projection of a knot map to the same surface (up to homeomorphism)? Neither? Both?
I'm having a bit of a hard time visualizing the schematics of an annulus.
How do we tell, looking at a "messy" surface, what are the boundaries and what are the holes? In Figure 4.44, how do we know that the surface has three boundary components and not four?