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Kauffman bracket skein modules of small 3-manifolds

Renaud Detcherry, Efstratia Kalfagianni, Adam S. Sikora

TL;DR

This work develops a concrete framework connecting Kauffman bracket skein modules S(M) to SL(2,ℂ) character varieties X(M) by exploiting tameness and reducedness. The authors establish dimension bounds and, under suitable hypotheses, exact equalities dim_{ℚ(A)} S(M) = |X(M)|, with a formulaic upper bound given by the dimension of the coordinate ring ℂ[X(M)]. They develop root-of-unity skein-module techniques via RT maps, construct equivariant actions, and prove reducedness criteria for Dehn-filled manifolds, including new computations for Dehn fillings of the figure-eight knot and (2,2n+1)-torus knots, yielding the first closed hyperbolic-manifold skein-module bases. They also show non-triviality of skein modules in broad classes and connect these results to Abouzaid–Manolescu invariants, offering a robust method to compute skein modules for large families of 3-manifolds and providing explicit bases via coordinate-ring structures. The findings significantly extend computability of skein modules and illuminate the algebraic-geometric structure underlying quantum invariants of 3-manifolds.

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$.

Kauffman bracket skein modules of small 3-manifolds

TL;DR

This work develops a concrete framework connecting Kauffman bracket skein modules S(M) to SL(2,ℂ) character varieties X(M) by exploiting tameness and reducedness. The authors establish dimension bounds and, under suitable hypotheses, exact equalities dim_{ℚ(A)} S(M) = |X(M)|, with a formulaic upper bound given by the dimension of the coordinate ring ℂ[X(M)]. They develop root-of-unity skein-module techniques via RT maps, construct equivariant actions, and prove reducedness criteria for Dehn-filled manifolds, including new computations for Dehn fillings of the figure-eight knot and (2,2n+1)-torus knots, yielding the first closed hyperbolic-manifold skein-module bases. They also show non-triviality of skein modules in broad classes and connect these results to Abouzaid–Manolescu invariants, offering a robust method to compute skein modules for large families of 3-manifolds and providing explicit bases via coordinate-ring structures. The findings significantly extend computability of skein modules and illuminate the algebraic-geometric structure underlying quantum invariants of 3-manifolds.

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed -manifolds are finitely generated over . In this paper, we develop a novel method for computing these skein modules. We show that if the skein module of is tame (e.g. finitely generated over ), and the -character variety is reduced, then the dimension is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension to the Abouzaid-Manolescu -Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed -manifolds and use it to compute the skein modules of Dehn fillings of -torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least over .
Paper Structure (29 sections, 42 theorems, 129 equations, 1 figure)

This paper contains 29 sections, 42 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tameness and reducedness under Dehn filling
  3. Skein modules of surgeries on $4_1$ and $T_{(2,2n+1)}$.
  4. Non-triviality of skein modules
  5. Outline of contents
  6. Skein modules at roots of unity
  7. Definition of the Kauffman bracket skein module
  8. Dimensions of skein modules at roots of unity.
  9. A construction of $\mathbb{C}[X(M)]$-equivariant maps on $S_{\zeta}(M)$
  10. Proof of Theorem \ref{['thm:inequality_dim']}
  11. On dimensions and bases of skein modules
  12. Examples with tame skein modules
  13. Reducedness of Character varieties
  14. Preliminaries
  15. Reducedness criteria
  16. ...and 14 more sections

Key Result

Theorem 1.1

If $M$ is a closed $3$-manifold with tame $S(M, \mathbb{Q}[A^{\pm 1}])$, then In particular, if $\mathcal{X}(M)$ is reduced, then $\dim _{\mathbb{Q}(A)}S(M)=|X(M)|$. Furthermore, $S(M, \mathbb{Q}[A^{\pm 1}])$ has no $(A+1)$-torsion in that case.

Figures (1)

  • Figure :

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • ...and 70 more