# A puzzle on knotted surfaces

Only after having learned that the somehow only notion of equivalence of knots is definitely "ambient isotopy" I stumbled over this blog entry on ambient isotopy. (Had it been earlier!)

What bothers me since then are the two puzzles at the bottom of this page: I have no glue how to solve them.

One of the puzzles asks to show that the following two embeddings of the genus two surface are equivalent:

Can anybody give me a hint? Or - even better - is there a movie around that shows the equivalence? (Or cannot there be such a movie - for specific reasons?)

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honestly you will just deprive yourself the fun of solving this.. just keep trying its possible, start at the RHS to get to the LHS. – user58512 Mar 1 '13 at 21:50
Why not start at the LHS to get to the LHS? Is this essential ("psychologically")? – Hans Stricker Mar 1 '13 at 21:52
obviously if it can be done it can be done backwards, but it comes much easier if you start from the RHS. – user58512 Mar 1 '13 at 21:53
My powers of imagination do not suffice, I'm afraid. Do you have another piece of assistance? – Hans Stricker Mar 1 '13 at 22:04
I couldn't resist thinking about the second one : i.stack.imgur.com/RKjvT.png still feels like witchcraft, though. – mercio Mar 2 '13 at 9:22

${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

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I appreciate that you hazard the consequences of being a spoiler! (And it still takes me some effort to digest your solution. It's not that easy than you might expect.) – Hans Stricker Mar 1 '13 at 22:28
To be honest: at first sight it looks like a "sleight of hand". I still cannot believe it. – Hans Stricker Mar 1 '13 at 22:31
@HansStricker, at least I can replace this with a problem that's 29x harder – user58512 Mar 1 '13 at 22:31
@HansStricker, You need to have Plasticine/Clay/Blue-Tac in your hands to really see this happen. – user58512 Mar 1 '13 at 22:32
I found it 29x harder than the usual ("simple") puzzles à la Thinking outside the box: en.wikipedia.org/wiki/Thinking_outside_the_box – Hans Stricker Mar 1 '13 at 22:36

I am glad my blog entry was useful. I agree that this is a matter of spoiling, but behind this link you can find the solutions to both of the problems in my blog post: http://wiki.helsinki.fi/download/attachments/72327738/toporatk1.pdf?version=2&modificationDate=1322829341390&api=v2 it is in Finnish, but I hopefully assume that your image-reading-Finnish is ok ;)

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Thanks a lot, Vadim. And welcome to MSE! – Hans Stricker Mar 4 '13 at 16:32