  A knot semigroup is defined by A. Vernitski. A. Vernitski conjectured that
the knot semigroup of the 2-bridge knot is isomorphic to an alternating sum
semigroup. To support this conjecture, and as a first main result, we prove
that the knot semigroup of the double twist knot is isomorphic to an
alternating sum semigroup. Moreover, as a second main result, we construct a
link invariant constructed by the Gelfand-Kiriliov dimension of algebra.
