The Jones polynomial is an invariant
of knots. This means that if two knots
are equivalent in the sense that
knot theory takes it to mean then
they have the same Jones polynomial.
You can compute the Jones polynomial
by making crossing changes until
the knot is trivial and keeping track
of the changes. The Jones polynomial was
defined as the trace of a braid corresponding to the knot in a certain representation of the braid group.