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Periodic Knots

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Josef Schröttle

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May 15, 2008, 10:02:23 AM5/15/08
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Hello,

how can I find the quotient knot to a given knot?

For example I got trefoil knot. Its quotient knot is the unknot, but how
can I figure that out?

Thanks for your help!
Josef

Lee Rudolph

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May 15, 2008, 2:15:01 PM5/15/08
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=?ISO-8859-15?Q?Josef_Schr=F6ttle?= <j...@bingo-ev.de> writes:


>Hello,
>
>how can I find the quotient knot to a given knot?
>
>For example I got trefoil knot. Its quotient knot is the unknot, but how
>can I figure that out?

What information do you assume you have about the cyclic group
action on S^3 that fixes your knot setwise? Do you, at a bare
minimum, know how your knot is situated with respect to the
fixed-point set of the action (if any)?

In the case of the trefoil K, if you know that it is placed
as a 2-string closed braid with axis A, and A is the fixed-point
set of an action of Z/3Z, and K is setwise fixed by that action,
then K is the closure of \sigma^3 (or \sigma^3), where \sigma is
a generator of the 2-string braid group, and in the quotient space
the quotient knot is the closure of \sigma, which is an unknot.

A similar argument could be made for any torus knot (and you could
bypass braid talk entirely, if you like), provided that your cyclic
action preserves the torus on which it lies. But of course there
are many more periodic knots than that.

What is the context in which your question arose?

Lee Rudolph

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