KBSM of lens spaces $L(p,2)$ and $L(4k,2k+1)$

TL;DR

The paper addresses the problem of explicitly computing and structurally describing the Kauffman bracket skein modules of lens spaces, focusing on $L(p,2)$ and $L(4k,2k+1)$ with $k\neq 0$, and the special case $L(0,1)$. It develops a two-fiber lens-space model $M_{2}(\beta_1,\beta_2)$ together with an arrow-diagram calculus on ${\bf S}^{2}$ to encode ambient isotopy, and leverages a basis for the KBSM of a fibered torus to construct new, computable bases for these skein modules. By introducing and exploiting polynomial families $\{P_m\}$, $\{Q_m\}$, and $\{P_{m,k}\}$, the authors build isomorphisms from diagrammatic modules to skein modules via maps $\varphi$ and $\psi$, yielding explicit bases $\Lambda_{\nu_1}$ and $\Sigma''_{\nu_1,\nu_2}$ and determining ranks and decompositions. In particular, they show $S_{2,\infty}(L(p,2);R,A) \cong R\Lambda_{\nu_1}$ and, for $L(4k,2k+1)$, $S\mathcal{D}_{\nu_1,\nu_2} \cong R\Sigma''_{\nu_1,\nu_2}$ with a rank of $2|\nu_0+1|+1$, while the case $\nu_0=-1$ (i.e., $L(0,1)$) yields infinite generation and a direct-sum decomposition into cyclic modules as $\mathcal{S}_{2,\infty}(L(0,1);R,A) \cong R \oplus \bigoplus_{i\ge 1} \frac{R}{(1-A^{2i+4})}$. The results extend classical work by Hoste–Przytycki and provide concrete, basis-driven computational tools for skein modules of key lens spaces, with potential implications for quantum invariants and 3-manifold topology.

Abstract

J. Hoste and J. H. Przytycki computed the Kauffman bracket skein module (KBSM) of lens spaces in their papers published in 1993 and 1995. Using a basis for the KBSM of a fibered torus, we construct new bases for the KBSMs of two families of lens spaces: $L(p,2)$ and $L(4k,2k+1)$ with $k\neq 0$. For KBSM of $L(0,1) = {\bf S}^{2}\times S^{1}$, we find a new generating set that yields its decomposition into a direct sum of cyclic modules.

Figures (20)

• Figure 1.1: Skein triple $L_{+}$, $L_{0}$, $L_{\infty}$ and $L\sqcup T_{1} + (A^{-2}+A^{2})L$
• Figure 2.1: Disk $\hat{\bf S}^{2}$ with marked points $\beta_{1}$ and $\beta_{2}$
• Figure 2.2: Arrow moves $\Omega_{1}-\Omega_{5}$ on ${\bf A}^{2}$
• Figure 2.3: $S_{\beta_{1}}$ and $S_{\beta_{2}}$-moves on $\hat{{\bf S}}^{2}$
• Figure 2.4: $\Omega_{\infty}$-move on $\hat{\bf S}^{2}$

Theorems & Definitions (59)

• Theorem 2.1
• Lemma 3.1: Lemma 4.3, DW2025
• Lemma 3.2: Lemma 4.4, DW2025
• Remark 3.3
• Lemma 3.4
• proof
• Remark 3.5
• Lemma 4.1
• proof
• Lemma 4.2