From: Ryan BUDNEY <budney@ihes.fr>
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the unpublished monolith of Bonahon and Siebenmann "Geometric splittings 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