  Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on
`Quantum knots and mosaics' to give a precise and workable definition of
quantum knots, intended to represent an actual physical quantum system. A knot
(m,n)-mosaic is an $m \! \times \! n$ matrix whose entries are eleven mosaic
tiles, representing a knot or a link by adjoining properly. In this paper we
introduce two variants of knot mosaics: period knot mosaics and toroidal knot
mosaics, which are common features in physics and mathematics. We present an
algorithm producing the exact enumeration of period knot (m,n)-mosaics for any
positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m
and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We
also analyze the asymptotics of the growth rates of their cardinality.
