  An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one
cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in
the boundary torus of $M$. For such a triangulation, the gluing and
completeness equations can be solved to recover the hyperbolic structure of
$M$. A planar projection of a knot gives four ideal cell decompositions of its
complement (minus 2 balls), two of which are ideal triangulations that use 4
(resp., 5) ideal tetrahedra per crossing. Our main result is that these ideal
triangulations are non-peripheral for all planar, reduced, alternating
projections of hyperbolic knots. Our proof uses the small cancellation
properties of the Dehn presentation of alternating knot groups, and an explicit
solution to their word and conjugacy problems. In particular, we describe a
planar complex that encodes all geodesic words that represent elements of the
peripheral subgroup of an alternating knot group. This gives a polynomial time
algorithm for checking if an element in an alternating knot group is
peripheral. Our motivation for this work comes from the Volume Conjecture for
knots.
