Kauffman bracket skein modules of small 3-manifolds

IF 1.5 1区 数学 Q1 MATHEMATICS
Renaud Detcherry , Efstratia Kalfagianni , Adam S. Sikora
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引用次数: 0

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed 3-manifolds are finitely generated over Q(A). In this paper, we develop a novel method for computing these skein modules.
We show that if the skein module S(M,Q[A±1]) of M is tame (e.g. finitely generated over Q[A±1]), and the SL(2,C)-character scheme is reduced, then the dimension dimQ(A)S(M,Q(A)) is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating dimQ(A)S(M,Q(A)) to the Abouzaid-Manolescu SL(2,C)-Floer theoretic invariants, for infinite families of 3-manifolds.
We prove a criterion for reducedness 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 Q(A).
小型3-流形的Kauffman支架绞合模块
Witten有限猜想的证明证明了闭3流形的Kauffman托架串模在Q(A)上是有限生成的。在本文中,我们开发了一种计算这些绞丝模的新方法。我们证明了如果M的串模S(M,Q[A±1])是驯服的(例如在Q[A±1]上有限生成),并且SL(2,C)-字符格式是约简的,那么维数dimQ(A) ^ S(M,Q(A))是该字符格式中的闭点数。特别地,这证实了文献中关于dimQ(a) S(M,Q(a))与Abouzaid-Manolescu SL(2,C)-花理论不变量的一个猜想。我们证明了闭3流形的特征变异的约简性判据,并利用它计算了(2,2n+1)环面结和8字形结的Dehn填充的串模。后一族给出了闭双曲型3流形串模的第一个计算实例。证明了有理同调球的绞结模的维数至少为1 / Q(A)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
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