{"id":1106,"date":"2014-03-28T22:12:50","date_gmt":"2014-03-28T22:12:50","guid":{"rendered":"http:\/\/blogs.oregonstate.edu\/glencora\/?p=1106"},"modified":"2014-03-28T22:12:50","modified_gmt":"2014-03-28T22:12:50","slug":"difference-hole-handle","status":"publish","type":"post","link":"https:\/\/blogs.oregonstate.edu\/glencora\/2014\/03\/28\/difference-hole-handle\/","title":{"rendered":"The difference between a hole and a handle"},"content":{"rendered":"

I have learnt topology in a very haphazard fashion. So sometimes when I observe something for the first time, my mind is blown. Topology is beautiful. Today’s lesson was in the difference between a hole and a handle.<\/p>\n

\"different-non-separating-cycles-on-surface\"<\/a><\/p>\n

So here’s the example, on the left are two examples of non-separating cycles on an orientable surface of genus 2.\u00a0 If we cut along them we get two surfaces that are topologically equivalent (right).\u00a0 To get back to the original surface, we can fill in the holes with disks (red), identify the two disks and then delete the disk.\u00a0 Call the two disks A and B.\u00a0 Since the surface on the right is orientable, when the disks are glued into the holes, there are two distinct sides of the disk, the inside and outside.\u00a0 There are three ways to glue these disks together:
\n(i) inside of A to inside of B
\n(ii) outside of A to outside of B
\n(iii) outside of A to inside of B or inside of A to outside of B
\nThe top example is (i) and the bottom is (ii).\u00a0 (iii) results in a non-orientable surface.\u00a0 Even though (i) and (ii) result in the same topological surface, the cycles used are fundamentally different because we have used a geometric idea of inside and outside.\u00a0 If I were living in the interior of this surface, the holes would look like handles and vice versa.\u00a0 [mind blowing sounds]\u00a0 In other words, given an orientable surface and a defined inside and outside a handle is a cycle that you can contract in the inside of the surface, and a hole is a cycle that you can contract in the outside of the surface.<\/p>\n","protected":false},"excerpt":{"rendered":"

I have learnt topology in a very haphazard fashion. So sometimes when I observe something for the first time, my mind is blown. Topology is beautiful. Today’s lesson was in the difference between a hole and a handle. So here’s the example, on the left are two examples of non-separating cycles on an orientable surface […]<\/p>\n","protected":false},"author":3747,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[106195],"class_list":["post-1106","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-topology"],"_links":{"self":[{"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/posts\/1106","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/users\/3747"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/comments?post=1106"}],"version-history":[{"count":1,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/posts\/1106\/revisions"}],"predecessor-version":[{"id":1111,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/posts\/1106\/revisions\/1111"}],"wp:attachment":[{"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/media?parent=1106"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/categories?post=1106"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/glencora\/wp-json\/wp\/v2\/tags?post=1106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}