Dear Ariadne,

Thank you for the link. Having just read the abstract, I think it will be very useful to me and I will take the time to study it carefully.

Niels

This is a good question.

It was Gordon who introduced n-knots $S^n \subset S^{n+2}$ not determined by their complements:

http://link.springer.com/article/10.1007/BF02568175#page-1

A good overview of higher-dimensional knot theory is given in

http://link.springer.com/article/10.1007/BF02568175#page-1

See Section 33, starting on page 453, on knot theory.   This book has an excellent bibliography.

Fortunately, the article by Gordon and Luecke is available at

http://www.ams.org/journals/bull/1989-20-01/S0273-0979-1989-15706-6/S0273-0979-1989-15706-6.pdf

It is pointed out that prime knots with isomorphic groups have

homeomorphic complements (p. 83).   This suggests that the answer to this question is positive, provided that the prime knots found have isomorphic groups.

A good overview of prime knots is given in

http://www.ams.org/journals/tran/1981-267-01/S0002-9947-1981-0621991-2/S0002-9947-1981-0621991-2.pdf

For intensive study of prime knots, see

http://math.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf

Dear James,

Thank you for the excellent response. I will take a look at the links you provided!

Best regards,

Niels

Thank you sir....

No -- thanks to an old paper by Cappell and Shaneson that declares this in its title (confusing the Hell out of anyone who doesn't read beyond the title to see that they're not talking about dimension three).