论文标题
随机步行和编织组的符号表示
Random walks and the symplectic representation of the braid group
论文作者
论文摘要
我们考虑$ sp（2l，\ mathbb {z}）$的编织组$ b（n）$的符号表示$ρ_n$，对于$ l = \ big [\ big [\ dfrac {n-1} {2} {2} {2} \ big] $。如果$ p $是$ 4L^2 $ polyenmial在$ sp（2l，\ mathbb {z}）$中的矩阵系数上，则表明b（n）中的$ \ {β\ in b（n）：p（ρ_n（β）= 0 \ \} $是$ ntanive Walks $ byenerate Walks $ n $ b（n）$ b（n）$。 如果$ n $很奇怪，我们得出的是，$ n $ braids $β$验证$ |δ_ {\hatβ}（ - 1）| \ 1）| \ leq c $对于某些常数$ c $形式的瞬态集：shere $Δ_ {\hatβ} $表示Alexander of $β$ $β$。 我们还推导出，对于随机$ 3 $编织，准蛋白链接$（βσ_iβ^{ - 1}σ_j）^p $对于每个整数$ p $和$ 1 \ leq I，j \ leq 2 $的签名为零。 \\作为这种辫子的一个例子，我们以$ 3 $ strands的形式调查了lissajous toric结的签名。
We consider the symplectic representation $ρ_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{Z})$, for $l=\Big[\dfrac{n-1}{2}\Big]$. If $P$ is a $4l^2$ polynomial on the coefficients of the matrices in $Sp(2l,\mathbb{Z})$, we show that the set $\{β\in B(n): P(ρ_n(β))=0\}$ is transient for non degenerate random walks on $B(n)$. If $n$ is odd, we derive that the $n$-braids $β$ verifying $|Δ_{\hatβ}(-1)|\leq C$ for some constant $C$ form a transient set: here $Δ_{\hatβ}$ denotes the Alexander polynomial of the closure of $β$. We also derive that for a random $3$-braid, the quasipositive links $(βσ_iβ^{-1}σ_j)^p$ have zero signature for every integer $p$ and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the signature of the Lissajous toric knots with $3$ strands.
