  Computing polynomial invariants for knots and links using braid
representations relies heavily on finding the trace of Hecke algebra elements.
There is no easy method known for computing the trace and hence it becomes
difficult to compute the known polynomial invariants of knots using their braid
representations. In this paper, we provide an algorithm to compute the trace of
the Hecke algebra representation of any braid. We simplify this algorithm and
write a Mathematica program to compute the invariants such as Alexander
polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology of a
very special family of knots and links $W(n,m)$ known as weaving knots by
expressing them as closure of weaving braids. We also explore on the
relationship between the topological and geometric invariants of this family of
alternating and hyperbolic knots (links) by generating data for the subfamilies
$W(3,m)$, $W(4,m)$, $W(5,m)$ and $W(6,m)$ of weaving knots.
