In Bigelow: Does the Jones Polynomial detect the Unknot?, J. Knot Theory Ramifications, 11, 493-505 (2002), Corollary 6.2. ff., a non-trivial braid $\beta$ in the kernel of the specialized Burau reperesentation (for $q=2$) of the Artin braid group $B_4$ is constructed. $\beta$ is given as a word in the standard generators $s_1$, $s_2$, $s_3$ of $B_4$. Let $H_4$ be the Hurwitz braid group, which is the Artin braid group $B_4$ with the additional relation $s_1s_2s_3s_3s_2s_1 = 1$ ($H_4$ is the spherical braid group $B_4(S^2)$).

The center $Z(H_4)$ of $H_4$ is a cyclic group of order $2$ generated by $(s_1s_2s_3)^4$. I need to verify that $\beta$ — now as an element of $H_4$ — is not contained in $Z(H_4)$. It would be sufficient to show that $\beta^2$ is not trivial in $H_4$.

Does a simple method exist, to decide, wether $\beta^2$ is not trivial in $H_4$?

Working with generators and relations or permutation representations of $H_4$ was not successful. A suitable matrix representation of $H_4$ could help. It is well known, that $H_4$ is linear, see for example Bardakov: Linear Representations of the Braid Groups of Some Manifolds, Acta Appl Math, 85, 41–48 (2005), Theorem 3.1, where an inclusion map of degree 144 is given.

Are there any (faithful) linear representations of $H_4$ with smaller degree?