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I am working with the Lawrence-Krammer representation of $B_n$ and need to find a way to determine if, given any two matrices $A, B$ in the image of the representation, there exists $k\in\mathbb{Z}^+$ such that $A = B^k$. I am working with the representation over $\mathbb{C}$ described in Budney $^\color{magenta}{\dagger}$ as it seemed to lend itself most naturally to this problem, but there are representations over $\mathbb{R}$ and $\mathbb{Z}[q^{\pm1}, t^{\pm1}]$. I tried checking eigenvalues for the matrices over $\mathbb{C}$, but the eigenvalues lie on the complex unit circle, so one would need to determine if there exists $m\in\mathbb{Z}^+$ such that the equation $k = \frac{u_1}{u_2} + 2\frac{m}{u_2}\in\mathbb{Z}^+$ some known $u_1, u_2\in\mathbb{R}$, which I am also struggling with. Is there any algorithm to determine if, given matrices $A, B$, there exists $k\in\mathbb{Z}^+$ such that $A = B^k$?

$^\color{magenta}{\dagger}$ Ryan D. Budney, On the image of the Lawrence-Krammer representation, Journal of Knot Theory and its Ramifications, Vol. 14, No. 06, pp. 773-789 (2005).

Gonçalo
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    The situation is tricky if you're looking for a numerical (i.e. computer based) method. For instance, if $B$ has distinct eigenvalues, with no eigenvalues equal to a root of unity, with no algebraic relation between the eigenvalues, and $A$ is any unitary matrix satisfying $AB = BA$, then there exists an increasing sequence of integers $k_1,k_2,\dots$ such that $B^{k_m} \to A$ as $m \to \infty$, even if $A = B^k$ has no solution – Ben Grossmann Mar 25 '25 at 20:50
  • This reminds me of the orbit problem – Rodrigo de Azevedo Mar 25 '25 at 21:38

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