In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know to be false by work of Bigelow, for instance).


**Intro.** The (reduced) Burau representation $\beta$ sends the Artin generators $\sigma\_i$ for the braid group $B\_n$ to the $(n-1)\times(n-1)$ matrix over the Laurent polynomials $\mathbb Z[t^\pm]$ given by
$$\beta(\sigma\_i) = \operatorname{Id}\_{i-2} \oplus \begin{pmatrix} 1 & 0 & 0 \\ t & -t & 1 \\ 0 & 0 & 1 \end{pmatrix} \oplus \operatorname{Id}\_{n-i-2}$$
with a slight, easy modification in the $i=1$ and $i=n-1$ cases when this formula doesn't make sense. The powers of $\sigma\_i$ have a nice formula that ends up being a bit cleaner to write if we substitute $t=-q$. We get
$$\beta(\sigma\_i^d) = \operatorname{Id}\_{i-2} \oplus \begin{pmatrix} 1 & 0 & 0 \\ -q(1+q+\dotsb+q^{d-1}) & q^d & (1+q+\dotsb+q^{d-1}) \\ 0 & 0 & 1 \end{pmatrix} \oplus \operatorname{Id}\_{n-i-2}$$


**Squier's conjecture.** In case $q$ "specialized" to be a primitive $d$-th root of unity, you can see that $\beta(\sigma\_i)$ has order $d$. Squier conjectured the following:



> 
> (C1) The kernel of the composite map $\beta\_{-q}:B\_n \xrightarrow{\beta} \operatorname{GL}\_{n-1}(\mathbb Z[t^\pm]) \xrightarrow{t \mapsto -q} \operatorname{GL}\_{n-1}(\mathbb C)$, when $q$ is a primitive $d$-th root of unity, is exactly equal to the normal subgroup $\langle\langle \sigma\_i^d \rangle\rangle$.
> 
> 
> 


**My issue.** Consider the full twist braid $\Delta^2=(\sigma\_1\dotsb\sigma\_{n-1})^n$ in $B\_n$. You can show that $\beta(\Delta^2)=t^n\cdot \operatorname{Id}\_{n-1}$. What power of $\Delta^2$ lies in $\ker \beta\_{-q}$? Consider the case $n=3$, $d=6$. Then $\beta(\Delta^2) = t^3 \mapsto -q^3 = 1$ under the specialization at $-q$, so $\Delta^2 \in \ker \beta\_{-q}$. But $\Delta^2 \notin \langle\langle \sigma\_i^6 \rangle\rangle$! (For instance, forgetting one strand sends eleemnts of $\langle\langle \sigma\_i^6 \rangle\rangle$ to 6-th powers in $B\_2$, but $\Delta^2$ maps to $\sigma\_i^2$.) It shouldn't be hard to find lots of examples like this, powers of $\Delta^2$ that easily lie in $\ker \beta\_{-q}$ but don't lie in $\langle\langle \sigma\_i^d \rangle\rangle$.


**Question.** Did Squier just overlook this seemingly simple counterexample to his conjecture? Does the conjecture seem more reasonable if we replace $\langle\langle \sigma\_i^d \rangle\rangle$ with $\langle \Delta^{2k} \rangle\langle\langle \sigma\_i^d \rangle\rangle$ for some power $k$ depending on $d$?


*Squier, Craig C.*, [**The Burau representation is unitary**](http://dx.doi.org/10.2307/2045338), Proc. Am. Math. Soc. 90, 199-202 (1984). [ZBL0542.20022](https://zbmath.org/?q=an:0542.20022).



Squier's conjecture on Burau at roots of unity
Presumably, this conjecture appears from the observation that the suitable powers of the standard generator $\sigma\_1$ lies in the kernel, although it looks that Squier missed other obvious elements of kernels as you mentioned.


I mention the paper by Funar and Kohno


On Burau’s representations at roots of unity.
<https://link.springer.com/article/10.1007/s10711-013-9847-0>


where they proved a strengthened form of another Squier's conjecture (C2) that appeared in the same Squier's paper. The paper also contains several discussions and results on Squire's conjectures.


It seems that a more suitable formulation of Squier's conjecture is as follows.


For $N=(n\_1,\ldots,n\_{k-1})$, let $B\_k\{N\}$ be the subgroup generated by  

$\sigma\_1^{2n\_1}, (\sigma\_1\sigma\_2)^{3n\_2}, \ldots, (\sigma\_1\sigma\_2\cdots \sigma\_{k-1})^{kn\_{k-1}}$.
Geometrically, the group $B\_k\{n\}$ is the group generated by $n\_i$-th powers of the Dehn twists along simple closed curves in $n$-punctured disk $D\_n$
enclosing $i+1$ puncture points.
When $N=\{n,n,n,\ldots\}$, we simply write $B\_k\{N\}$ by $B\_k\{n\}$.


**Modified Squier's conjecture, weak version**
If $q$ is a primitive root of unity then the kernel of Burau representation $\beta\_{-q}$ is equal to $B\_k\{N\}$ for suitable $N$.


**Modified Squier's conjecture, strong version**
If $q$ is a primitive root of unity then the kernel of Burau representation $\beta\_{-q}$ is equal to $B\_k\{n\}$ for some $n$.


If the Modified conjecture (strong version) is true for infinitely many distinct primitive root of unity, then the Burau representation is faithful.
This follows from Theorem 2.1 of Funar-Kohno's paper that asserts that the intersection of $B\_k\{n\}$ over an infinite set of integers $n$ is trivial.
Thus for $k>4$, the Modified conjecture (strong version) can be true for only finitely many primitive root of unities.


On the other hand, Corollary 3.2 of Funar-Kohno's paper shows that if $q$ is a primitive $2n$-th root of unity for odd $n$ and $2n\geq7$, then modified conjecture (strong version) is true:
Corollary 3.2 says that the kernel $K$ of the Burau representation of 3-braids at $-q$ is normally generated by $\sigma\_1^{2n}, \sigma\_2^{2n} ,(\sigma\_1^{2}\sigma\_2^{2})^{n}$. Since
$ (\sigma\_1\sigma\_2)^{3n}=(\sigma\_1^{2}(\sigma\_2\sigma\_1^{2}\sigma\_2))^{n} = \sigma\_1^{2n} (\sigma\_2 \sigma\_1^{2}\sigma\_2)^{n}$
and $(\sigma\_2 \sigma\_1^{2}\sigma\_2)^{n}$ is conjugate to $(\sigma\_1^{2}\sigma\_2^{2})^{n}$, the kernel $K$ coincides with $B\_3\{n\}$.


These results say that the modified conjecture (weak or strong version) is meaningful and subtle for $k=4$ case. I feel it reasonable to investigate the modified conjectures as a possible approach of the faithfulness problem of Burau representation of 4-braids.


