Sliced skein algebras and geometric Kauffman bracket

Abstract

The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=Σ_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_ξ(\mathfrak{S})$ corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter $ξ$ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is a fully Azumaya point of the skein algebra $\mathcal{S}_ξ(\mathfrak{S})$. For any $SL_2(\mathbb{C})$--representation $ρ$ of the fundamental group of an oriented connected 3-manifold $M$ and a root of unity $ξ$ with odd $ord(ξ^2)$, we introduce the $ρ$-reduced skein module $\mathcal{S}_{ξ,ρ}(M)$. We show that $\mathcal{S}_{ξ,ρ}(M)$ has dimension 1 when $M$ is closed and $ρ$ is irreducible. We also show that if $ρ$ is irreducible the $ρ$-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.

Figures (4)

• Figure 1: From left to right: ${\mathbb P}_3, {\mathbb P}_2, {\mathbb P}_1$. The $Y$ graph is in red.
• Figure 2: Standard curves $\ell_1, a_{23}, a_{11}$.
• Figure 3: Left: $t$-slide, Right: loop-slide. Here $c\in \mathcal{C}$.
• Figure 4: The DT-datum $(\mathcal{C},\Gamma)$, with $\Gamma$ in red. Left: the case $g\ge 1$. Right: $g=0$. The small white disks are the punctures.

Theorems & Definitions (118)

• Theorem 1: See Theorem \ref{['thmMain2']}
• Corollary 2
• Theorem 3: See Theorem \ref{['thmMain2']}
• Theorem 4: See Theorem \ref{['thmCenter']}
• Theorem 5: Theorem \ref{['thmSliAzu']}
• Remark 1.1
• Theorem 6: See Theorem \ref{['thmAzuRep']}
• Theorem 7: See Theorem \ref{['thmGeoKau']}
• Remark 1.2
• Lemma 2.1