  We present a new and very concrete connection between cluster algebras and
knot theory. This connection is being made via continued fractions and snake
graphs. It is known that the class of 2-bridge knots and links is parametrized
by continued fractions, and it has recently been shown that one can associate
to each continued fraction a snake graph, and hence a cluster variable in a
cluster algebra. We show that up to normalization by the leading term the Jones
polynomial of the 2-bridge link is equal to the specialization of this cluster
variable obtained by setting all initial cluster variables to 1 and
specializing the initial principal coefficients of the cluster algebra as
follows $y_1=t^{-2}$ and $ y_i=-t^{-1}$, for all $i> 1$. As a consequence we
obtain a direct formula for the Jones polynomial of a 2-bridge link as the
numerator of a continued fraction of Laurent polynomials in $q=-t^{-1}$. We
also obtain formulas for the degree and the width of the Jones polynomial, as
well as for the first three and the last three coefficients. Along the way, we
also develop some basic facts about even continued fractions and construct
their snake graphs. We show that the snake graph of an even continued fraction
is ismorphic to the snake graph of a positive continued fraction if the
continued fractions have the same value. We also give recursive formulas for
the Jones polynomials.
