Let $q\in \mathbb{C}$ and $\mathcal{A}_{q}$ be a Hecke algebra of degree $n$, i.e. a unital algebra generated by elements $\sigma_1,...,\sigma_{n-1}$ satisfying the following relations
$\sigma_{k}\sigma_{k+1}\sigma_{k}=\sigma_{k+1}\sigma_{k}\sigma_{k+1}$ , for $1\le k \le n-2$
$\left(\sigma_k-1\right)\left(\sigma_k+q^21\right)=0$
$\sigma_k\sigma_l=\sigma_l\sigma_k$ for $|k-l|\ge 2$.
Suppose we have a permutation $\pi\in \Pi_{n}$ and let $t_j$ be a transposition $(j,j+1)$. Let $\pi=t_{k_1}...t_{k_l}$ be a decomposition of $\pi$ into a minimal number of transpositions. How to prove that $\sigma_{k_1}...\sigma_{k_l}$ doesn't depend on the actual choice of transpositions as far as their number is minimal ?
In the symmetric group, a result commonly called "Matsumoto Lemma" allows you to pass from any reduced expression of a permutation to any other using only braid relations, that is, the first and last relations which you gave. Since these relations also hold in the Hecke algebra, any sequence of braid relations applied to a reduced expression of a permutation can be applied to the corresponding lifted reduced expression in the Hecke algebra. Hence you obtain, by a sequence of braid relations, a way to pass from a lifted reduced expression to another exactly as in the symmetric group.