From: Ryan BUDNEY
Date: Tue, 7 Nov 2006 00:04:48 +0100 (CET)
Here is a response to Joan Birman's question.
Dear Joan,
Schubert's paper is from before the time when knot theory and 3-manifold
theory were united so it is rather akward reading with much
no-longer-in-use terminology. I put a paper on the arXiv
"JSJ-decompositions of knot and link complements in S3" (to appear in
l'Enseignement Mathematique) which gives a uniqueness theorem for
satellite knots, reproving and generalizing Schubert's theorems on
satellite/companion knots and links in the process, via the JSJ
decomposition + Hyperbolisation theorem. I do not get into any of the
other topics of "Knoten und Vollringe" as I just wanted to have a more or
less elementary account of the basic results on satellite/companions from
a modern point of view. The results in my paper were also in part proven
by Sakuma in "Uniqueness of symmetries of knots" Math.Z. 192, and also in
the unpublished monolith of Bonahon and Siebenmann "Geometric splittings
of knots and Conway's algebraic knots". It would be nice if more of
Schubert's work was "exposed" in this way as his work seemingly `set the
stage' for the JSJ-decomposition.
A question of a similar spirit to put to the list: Other than Seifert and
Threlfall, are there any introductions to algebraic topology that cover
the torsion linking forms on compact orientable manifolds?
For example, given a compact oriented 3-manifold M, there is a form with
values in Q/Z, defined on the torsion subgroup of H_1(M;Z), induced
essentially from Poincare duality. This form allows for a rather
quick-and-easy homotopy classification of 3-dimensional lens spaces.
-ryan budney