Let $\pi_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define a natural transformation $\eta: \pi_{1} \rightarrow H_{1}$ with a component $\eta_{X}: \pi_{1}(X) \rightarrow H_{1}(X)$. We define the configuration space on $n$ points by, $$C_{\hat{n}}(\mathbb{R}) = \{(z_{1},...,z_{n}) \in \mathbb{R}^n \; | \; z_{i} \neq z_{j}\}$$ and the unordered configuration space on $n$ points by $C_{n}(\mathbb{R}) = C_{\hat{n}}(\mathbb{R}) \big/ \Sigma_{n}$, where $\Sigma_{n}$ is the symmetric group. We then have for $\vec{p} \in C_{\hat{n}}(\mathbb{R})$ that $\pi_{1}(C_{\hat{n}}(\mathbb{R}),\vec{p}) \cong PB_{n}$ and $\pi_{1}(C_{n}(\mathbb{R}),\tau(\vec{p})) \cong B_{n}$, where $\tau : C_{\hat{n}}(\mathbb{R}) \rightarrow C_{n}(\mathbb{R})$ is the orbit space projection, $PB_{n}$ is the pure braid group on $n$-strands, and $B_{n}$ is the braid group on $n$-strands.

Consider the components of the natural transformation between the fundamental group functor and the first homology group functor given by,

$$\eta_{C_{\hat{n}}(\mathbb{R})} : \pi_{1}(C_{\hat{n}}(\mathbb{R})) \rightarrow H_{1}(C_{\hat{n}}(\mathbb{R}))$$ $$\eta_{C_{n}(\mathbb{R})} : \pi_{1}(C_{n}(\mathbb{R})) \rightarrow H_{1}(C_{n}(\mathbb{R}))$$

What objects in $\text{Grp}$ will these components of $\eta: \pi_{1} \rightarrow H_{1}$ associate with $PB_{n}$ and $B_{n}$, and how do they relate to $C_{\hat{n}}(\mathbb{R})$ and $C_{n}(\mathbb{R})$?

I am interested in this question because I am wondering if $H_{1}(C_{\hat{n}}(\mathbb{R}))$ and $H_{1}(C_{n}(\mathbb{R}))$ have any unnoticed connections to knot theory since every knot is the closure of a braid.