The set of fixed points which do not move as a knot is transformed into itself is not a knot. The conjecture was proved in
1978 (Morgan and Bass 1984). According to Morgan and Bass (1984), the Smith conjecture
stands in the first rank of mathematical problems when measured by the amount and
depth of new mathematics required to solve it.

The generalized Smith conjecture considers to be a piecewise linear -dimensional hypersphere
in ,
and
the -fold
cyclic covering of
branched along ,
and asks if
is unknotted if
is an
(Hartley 1983). This conjecture is true for , and false for , with counterexamples in the latter case provided by
Giffen (1966), Gordon (1974), and Sumners (1975).

Giffen, C. H. "The Generalized Smith Conjecture." Amer. J. Math.88, 187-198, 1966.Gordon, C. M. "On
the Higher-Dimensional Smith Conjecture." Proc. London Math. Soc.29,
98-110, 1974.Hartley, R. "Whitehead Torsion and the Smith Conjecture."
Michigan Math. J.30, 121-128, 1983.Morgan, J. W.
and Bass, H. (Eds.). The
Smith Conjecture, Papers Presented at the Symposium Held at Columbia University,
New York, 1979. Orlando, FL: Academic Press, 1984.Rolfsen, D.
Knots
and Links. Wilmington, DE: Publish or Perish Press, pp. 350-351, 1976.Smith,
P. A. "Transformations of Finite Period. II." Ann. Math.40,
690-711, 1939.Summers, D. W. "Smooth Actions on Spheres which Leave Knots Pointwise Fixed."
Trans. Amer. Math. Soc.205, 193-203, 1975.Waldhausen,
F. "Über Involutionen der 3-Sphäre." Topology8,
81-91, 1969.