H. Gruber showed the following results toward a solution of this problem.
- If the statement of this problem is true for all prime knots, then it
is true for all knots.
- The statement is true for all alternating knots whose Conway basic
polyhedron has odd chromatic invariant. In particular, the statement is
true for all algebraic alternating knots.
See [Gruber, Math. Proc. Camb. Phil. Soc. 147 (2009)].
M. Eisermann showed that this conjecture is true.
Since h_X(K) is bounded by |X|^(braid index of K),
the conjecture is obtained from Theorem 3 of
Eisermann, Proc. Amer. Math. Soc. 128 (2000)].
This problem has been solved by Budney.
The homotopy type of the space of long knots
with a fixed knot type
can be calculated by reducing it
to the homotopy types of the spaces of long knots
with shorter companionship trees;
comments or suggestions to problems of the problem list would be welcome.