Kauffman bracket skein module of two families of Seifert manifolds

TL;DR

The article computes the Kauffman bracket skein modules of two families of Seifert manifolds by deriving explicit presentations and analyzing handle-sliding submodules. It proves that S_{2,∞}(D^2(k_1,k_2)) is an infinitely generated free module for k_i ≥ 1, while S_{2,∞}(S^2(k_1,k_2,k_3)) is finitely generated for k_i ≥ 2, with the minimal generator count bounded by (k_1+1)(k_2+1)(k_3+1). The methods combine surgery-diagram techniques, Chebyshev decorations, and detailed relator analysis to isolate explicit bases G^{k_1,k_2} and G^{k_1,k_2,k_3}, and the results include the nontriviality of the empty link in both settings. This work advances understanding of skein modules for small Seifert manifolds and provides constructive presentations that can guide future investigations of related 3-manifolds.

Abstract

We compute the Kauffman bracket skein modules of Seifert manifolds $Σ_{0,1}((k_1,1),(k_2,1))$ and $Σ_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of $Σ_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\ge 1$ and that of $Σ_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \ge 2$. We also show that the empty link in either case is not trivial.

Figures (16)

• Figure 2.1: Skein triple for the Kauffman bracket skein module.
• Figure 2.2: $S_{l_1}(a_1)S_{l_2}(a_2)S_{l_3}(a_3)$ and $S_{l}(a_2)C_{m,n}$
• Figure 2.3: Heegaard splitting and surgery diagram of $S^2(k_1,k_2,k_3)$
• Figure 2.4: Heegaard surface in surgery diagram of $S^2(k_1,k_2,k_3)$
• Figure 3.1: $R(n_1,n_2)$(in this figure, $n_1<0$, $n_2>0$)

Theorems & Definitions (96)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: 1998Fundamentals
• Remark 2.5
• Corollary 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9