Alexander polynomials for a certain family of closed braids

Question

Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\dots \sigma_n^{k_n}$, where $\sigma_i=\sigma_i^+$ is the generator taking the $i$-th string to the $(i+1)$-st string and $(i+1)$-st string to the $i$-th string in a single overlap (as in Fig. 1.9 on p. 16 of Kassel and Turaev, Braid groups, volume 247, Springer science and Business media).

On closing up the braid $\sigma_\kappa$ one obtains a link, which we call $L_\kappa$, with associated Alexander polynomial $\Delta_\kappa(t)\in \Lambda$, where $\Lambda=\mathbb{Z}[t^{\pm 1}]$, which is well defined up to a unit in $\Lambda$. A well known formula for the Alexander polynomial $\Delta_b(t)$ of a link associated to a braid $b$ states that $\Delta_{b}(t)=\frac{(1-t)}{(1-t^n)}\operatorname{det}\left(\operatorname{I}_n-\bar{\psi}_n(b)\right)$, where $\bar{\psi}_n: B_n\rightarrow \operatorname{GL}_{n-1}(\Lambda)$ is the reduced Burau representation (c.f. Chapter 3 of loc. cit.), which arises from interpreting $B_n$ as the mapping class group of a disk with $n$ marked points.

Using this formula, I was able to show for $n=3, 4$ that the Alexander polynomial $\Delta_\kappa(t)$ is $\prod_{i=1}^n F_{k_i}(t)$, where $F_r(t):=1-t+t^2+\dots+(-1)^{r-1}t^{r-1}=\frac{1-(-1)^r t^r}{1+t}$. When $n$ gets larger, the size of the matrices get larger and in general it is not clear to me if a formula similar to this should hold for all $n$? I wasn't able to find this computation appear in the literature, but it seems like something that should be known. Also, I was wondering if this family of braids has any special significance.

Answer 1

The closure of the braid $\sigma_\kappa$ is a connected sum of torus links $T(2,k_i)$ (which are closures of 2-braids). Since the Alexander polynomial is multiplicative with respect to connected sums, $\Delta_\kappa = \prod_i \Delta_{T(2,k_i)}$.

You can see more generally that if you have a braid $\beta = \beta_{\rm top}\beta_{\rm bot} \in B_n$ which is a product of two braids where $\beta_{\rm top} \in \langle \sigma_1,\dots,\sigma_m\rangle$ and $\beta_{\rm bot}\in \langle \sigma_{m+1},\dots,\sigma_{n-1}\rangle$, then the closure of $\beta$ is the connected sum of the closures of $\beta_{\rm top}$ and $\beta_{\rm bot}$ (viewed as an $(m+1)$-braid and an $(n-m)$-braid, respectively). The circle running (mostly) between the $(m+1)^{\rm st}$ and the $(m+2)^{\rm nd}$ strands exhibits the connected sum.