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?

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    $\begingroup$ It seems that the body of your post is about one question, but the title and the final question are only about one possible approach to the proof of that question. Why not use your real question as the title? \\ Also, please use MathJax rather than Markdown to format your answers; it's very hard to read otherwise. I have edited accordingly. $\endgroup$ – LSpice Nov 14 '20 at 20:31

A group is said to have solvable word problem if there is an algorithm allowing one to determine in finite time if a word represents the identity or not. Of course, if you have a faithful linear representation of your group, then the word problem is solvable as checking whether a word represents the trivial element or not amounts to checking that a matrix product is the identity. However, such solutions to the word problem are not always easy to implement (depending for instance on the base field or ring on which the representation is defined) and have complexity which is far from being optimal in general.

However, in the case of surface braid groups (including $H_4\cong B_4(S^2)$), the word problem is known to be solvable, using combinatorial algorithms not relying on a linear representation: this implies that you have an algorithm allowing you to check whether $\beta^2=1$ or not. The case of $B_n(S^2)$ is given in the very last theorem of the following paper : Edward Fadell and James Van Buskirk, The braid groups of $E^2$ and $S^2$, Duke Math. J., Volume 29, Number 2 (1962), 243-257. It gives a normal form for every element of $B_n(S^2)$, $n\geq 4$. For explicit algorithms you might look at the last section of Ruan González-Meneses' paper New presentations of surface braid groups, J. Knot Theory Ramifications 10 (2001), no. 3, 431–451 (also available on arxiv : https://arxiv.org/abs/math/9910020 ).

(Note that González-Meneses' paper gives new presentations for braid groups of closed surfaces different from the sphere and projective plane; however as mentioned in the introduction, the obtained presentations are similar to those of the above mentioned paper of Fadell and Van Buskirk and the algorithm described in the last section also applies to these groups).


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