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According to C.C. Adams, The knot book, pp 224, "every knot is either pass equivalent to the trefoil knot or the unknot".

A pass move is the following: enter image description here

Can someone show me how to show that the Cinquefoil knot is pass equivalent to unknot or trefoil? Been trying on paper but no luck. Don't see how pass moves apply here. Thanks.

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Using Reidemeister moves would also be ok, have also tried that, also no success. – Rudy the Reindeer Sep 6 '12 at 12:28
up vote 1 down vote accepted

The general method is demonstrated in Kauffman's book On Knots. Put the knot into a "band position" So that the Seifert surface is illustrated as a disk with twisted and intertangled bands attached. Then the orientations match those of your figure. You can pass one band over another. Your knot is the braid closure of $\sigma_1^5$. The Seifert surface is two disks with 5 twisted bands between them. Start by stretching the disks apart.

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Thank you! You're my first answerer since I started to learn knot theory here on SE : ) Quite happy to have found you. I have uploaded a picture of my Seifert surface for the cinquefoil knot in one of my previous questions here. So, just to be sure, I should have kept the orientation in my third drawing then I'd've seen that I can for example swap the two non-twisted bands between the disks? – Rudy the Reindeer Sep 6 '12 at 12:45
Yes, the diagram of the Seifert surface that you have should do the trick. I haven't worked this specific example though. – Scott Carter Sep 6 '12 at 14:31
Also, I am quite busy this morning, so I won't check the details of your computations. Usually, it takes me at least 3 tries to get these things correct :/ – Scott Carter Sep 6 '12 at 14:33
Yes, I seem to struggle to get it right, too : / – Rudy the Reindeer Sep 6 '12 at 17:23
I put a bounty on my other question. – Rudy the Reindeer Sep 8 '12 at 10:56

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