  We introduce tensor network contraction algorithms for the evaluation of the
Jones polynomial of arbitrary knots. The value of the Jones polynomial of a
knot maps to the partition function of a $q$-state Potts model defined as a
planar graph with weighted edges that corresponds to the knot. For any integer
$q$, we cast this partition function into tensor network form and employ fast
tensor network contraction protocols to obtain the exact tensor trace, and thus
the value of the Jones polynomial. By sampling random knots via a grid-walk
procedure and computing the full tensor trace, we demonstrate numerically that
the Jones polynomial can be evaluated in time that scales subexponentially with
the number of crossings in the typical case. This allows us to evaluate the
Jones polynomial of knots that are too complex to be treated with other
available methods. Our results establish tensor network methods as a practical
tool for the study of knots.
