From: Daniel Ruberman
Subject: Re: Embeddings
Date: Tue, 15 Jun 1999 13:28:41 -0400
Newsgroups: [missing]
To: Dave Rusin
Keywords: Distinct links with homeomorphic complements in S^3
Dear Dave--The examples regarding links are simple, and can be described as
follows. Imagine a two (or more component) link, in which one component,
say K, is unknotted and so spans a disk D. To make the example
interesting, suppose that the rest of the components hit D in at least 2
points. Now cut the complement along D, and reglue with a twist of 2\pi (a
"Dehn twist"). This is a homeomorphism of the complement to itself. The
image of the other components, together with K, gives a new link in the
3-sphere, which is presumably not equivalent to the original link. (The
homeomorphism of the complement takes the meridian of K to the sum of a
meridian and longitude, and so doesn't extend over S^3). More or less any
link of this kind will work; the simplest is the "Whitehead link", where
the second component goes through D twice, but clasps itself. (YOu can
tell the links are different by looking at the Alexander polynomial; I
think this is in Rolfsen's book "Knots and links". If you twist by 4pi,
the second component becomes knotted, so the resulting link is certainly
not equivalent to the original.)
Daniel Ruberman
>Thank you for supplying pointers. Do you also have a handy reference for the
>negative result about links? (I was sure I had heard this result but
>couldn't provide the original poster with any useful citation.)
>
>dave
==============================================================================
From: Daz
Subject: Re: Embeddings
Date: Wed, 16 Jun 1999 02:25:05 -0700 (PDT)
Newsgroups: [missing]
To: rusin@math.niu.edu (Dave Rusin)
>
> >but is false for links in R^3.
> I also mentioned this to the poster but had no clue where to find a reference.
> Got any leads?
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I don't know a reference but I know a simple example: Consider the Whitehead
link arranged so that 1 component is a geometric circle, and the other one
is -- well you know, the usual. Think of the second component as embedded in
a solid torus T. Now introduce an arbitrary number n of 360-deg.
twists into the second component by imposing a self-homeomorphism T -> T
that restricts to bd(T) as n successive Dehn twists. (If any of this is
unclear, I apologize; if sketching were easy to send via e-mail it would be
a lot easier to convey.)
It is easy to show that the topological type of the complement of this link
= link_n is independent of n, but the link type depends on n.
(I heard this from Bill Thurston.)
--Dan Asimov