Difference between revisions of "Notes:Potentially invalid vertex scheme"
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*** But in the case described the {{C|ada}}-to-{{C|ada}} join is not ambiguous, dealing with that first we see that it is very much so. Do we twist or not? | *** But in the case described the {{C|ada}}-to-{{C|ada}} join is not ambiguous, dealing with that first we see that it is very much so. Do we twist or not? | ||
{{Caveat|Is this even a "valid" vertex labelling scheme?}} | {{Caveat|Is this even a "valid" vertex labelling scheme?}} | ||
+ | * After playing around with a leather bookmark, if you do the {{C|ada}}-to-{{C|ada}} join with a twist in it, thus yielding a mobius band AND then try and fold it... you cannot. | ||
'''Problems:''' | '''Problems:''' | ||
# The left and right edges, {{c|ada}} are ambiguous, do we twist or not (mobius band vs tube) | # The left and right edges, {{c|ada}} are ambiguous, do we twist or not (mobius band vs tube) |
Latest revision as of 03:36, 31 January 2017
Contents
Instance
This is done in the context of Munkres - Elements of Algebraic Topology - page 16:
- What is the geometric realisation of the complex shown to the right?
- Surely [ilmath]\cong\mathbb{S}^1\times I[/ilmath] where [ilmath]I:\eq[0,1]\subset\mathbb{R} [/ilmath]?
- intuitively, if one folds along defd to join abca to itself, we get a rectangle. We then have to bend it round to join ada to ada, thus making a tube.
- However, we do have to identify be to eb and ditto for cf... I'm not sure what these do
- But in the case described the ada-to-ada join is not ambiguous, dealing with that first we see that it is very much so. Do we twist or not?
Caveat:Is this even a "valid" vertex labelling scheme?
- After playing around with a leather bookmark, if you do the ada-to-ada join with a twist in it, thus yielding a mobius band AND then try and fold it... you cannot.
Problems:
- The left and right edges, ada are ambiguous, do we twist or not (mobius band vs tube)
- be-to-be join
- cf-to-cf join
"Solution"
- This should perform "more identifications than one would think" and not be a torus.