Given a braid presentation
of a hyperbolic knot, Hikami and Inoue consider a system of polynomial
equations arising from a sequence of cluster mutations determined by
. They show
that any solution gives rise to shape parameters and thus determines a boundary-parabolic
–representation
of the knot group. They conjecture the existence of a solution corresponding to the
geometric representation. Here we show that a boundary-parabolic representation
arises from a solution if and only if the length of
modulo
equals the obstruction to
lifting
to a boundary-parabolic
–representation (as
an element in
).
In particular, the Hikami–Inoue conjecture holds if and only if the length of
is
odd. This can always be achieved by adding a kink to the braid if necessary. We
also explicitly construct the solution corresponding to a boundary-parabolic
representation given in the Wirtinger presentation of the knot group.
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