Let n be any natural number. Let K be any n-dimensional knot in
Sn+2. We define a supersymmetric quantum system for K with the following
properties. We firstly construct a set of functional spaces (spaces of
fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric
infinitesimal transformations) in an explicit way. Thus we obtain a set of the
Witten indexes for K. Our Witten indexes are topological invariants for
n-dimensional knots. Our Witten indexes are not zero in general. If K is
equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten
indexes restrict the Alexander polynomials of n-knots. If one of our Witten
indexes for an n-knot K is nonzero, then one of the Alexander polynomials
of K is nontrivial. Our Witten indexes are connected with homology with
twisted coefficients. Roughly speaking, our Witten indexes have path integral
representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure