We consider the symplectic representation $\rho_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{ Z})$, for $l=\Big[\dfrac{ n-1}{ 2}\Big]$. If $P$ is a polynomial on the $4l^2$ coefficients of the matrices in $Sp(2l,\mathbb{ Z})$, we show that the set $\{ \beta\in B(n): P(\rho_n(\beta))=0\}$ is transient for non degenerate random walks on $B(n)$. We derive that the $n$-braids $\beta$ which close into a loop $\hat{ \beta}$ with $0<|det({ \hat{ \beta}})|\leq C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than $\dfrac{ 1}{ p}$. We also derive that for a random $3$-braid, the quasipositive links $(\beta\sigma_i\beta^{ -1}\sigma_j)^p$ have zero signature for every integer $p$ and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the signature of the Lissajous toric knots $3$-braids.
Categories: math.GT