The pair consisting
of a knot and a
surjective map
from the knot group onto a dihedral group of order
for
an odd integer is
said to be a –coloredknot. In [Algebr. Geom. Topol. 6 (2006) 673–697] D Moskovich conjectures that there are exactly
equivalence
classes of –colored
knots up to surgery along unknots in the kernel of the coloring. He shows that for
and
the conjecture holds
and that for any odd
there are at least
distinct classes, but gives no general upper bound. We show that there are at most
equivalence classes
for any odd .
In [Math. Proc. Cambridge Philos. Soc. 131 (2001) 97–127] T Cochran, A Gerges
and K Orr, define invariants of the surgery equivalence class of a closed
–manifold
in the context of
bordism. By taking
to be –framed
surgery of
along
we may define Moskovich’s colored untying invariant in the same way as the
Cochran–Gerges–Orr invariants. This bordism definition of the colored untying invariant
will be then used to establish the upper bound as well as to obtain a complete invariant of
–colored
knot surgery equivalence.