Talk:Crosscap number
Add topicLatest comment: 4 years ago by IsaacSundbe in topic Error in example?
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Cross-cap
[edit]How does the cross-cap fit in with this? I seem to recal the Euler characteristic is related to the number of cross-caps but the details escape me. --Salix alba (talk) 19:54, 24 September 2006 (UTC)
- If you consider the classification of surfaces, for compact non-orientable surfaces, the classification states for a given Euler characteristic and specified number of boundary components, we have connected sum of a determined number of projective planes minus discs (to get the boundary components). "Cross-cap" can loosely refer to these summands. So the definition currently given is equivalent to counting the number of such crosscaps. --C S (Talk) 23:54, 9 March 2007 (UTC)
Error in example?
[edit]The article says that the crosscap number of the unknot is zero, because it bounds a disk (of Euler Characteristic 1). The disk is orientable, so this doesn't fit into the definition provided. I would have said the crosscap number is 1 because the unknot bounds a punctured RP^2 (of Euler characteristic 0). This is my first time encountering crosscap numbers, so I was not comfortable making the change (in case I have missed something). — Preceding unsigned comment added by IsaacSundbe (talk • contribs) 14:33, 9 February 2022 (UTC)