Random walks and the symplectic representation of the braid group
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915468754812928 |
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| author | Soret, Marc Ville, Marina |
| author_facet | Soret, Marc Ville, Marina |
| contents | 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 polynomial on the $4l^2$ 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 $B(n)$.
We derive that the $n$-braids $β$ which close into a loop $\hatβ$ with $0<|det({\hatβ})|\leq C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than $\dfrac{1}{p}$.
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 $3$-braids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_00984 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Random walks and the symplectic representation of the braid group Soret, Marc Ville, Marina Geometric Topology 57K10, 57K14, 60B15, 60B20, 60G50 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 polynomial on the $4l^2$ 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 $B(n)$. We derive that the $n$-braids $β$ which close into a loop $\hatβ$ with $0<|det({\hatβ})|\leq C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than $\dfrac{1}{p}$. 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 $3$-braids. |
| title | Random walks and the symplectic representation of the braid group |
| topic | Geometric Topology 57K10, 57K14, 60B15, 60B20, 60G50 |
| url | https://arxiv.org/abs/2203.00984 |