![](/files/happy5.png)
Unsolvability of word problem for 2-knot complements
Problem Does there exist a smooth/PL embedding of
in
such that the fundamental group of the complement has an unsolvable word problem?
![$ S^2 $](/files/tex/1cd459995f11529f346339e6879cf139c22ee92c.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
It's known that there are smooth -dimensional submanifolds of
whose fundamental groups have unsolvable word problems. The complements of classical knots (
) are known to have solvable word problems, as do arbitrary
-manifold groups.
Bibliography
A. Dranisnikov, D. Repovs, "Embeddings up to homotopy type in Euclidean Space" Bull. Austral. Math. Soc (1993).
* indicates original appearance(s) of problem.