Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3‐manifolds
Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3‐manifolds
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一些简单 3 流形的 Witten-Reshetikhin-Turaev 不变量的渐近展开
DOI：
10.1063/1.531237
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发表时间：
1995
影响因子：
1.3
通讯作者：
R. Lawrence
中科院分区：
文献类型：
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作者：
R. Lawrence
对于任何李代数g和整数水平k，定义了一个3-流形M中链L的嵌入不变量Zk*（M，L），称为Witten-Reshetikhin-Turaev不变量。已知对于S3中的链路，Zk*（S3，L）是q=exp（2πi/（k+cgv））中的多项式，即链路L的广义Jones多项式。本文研究了一个简单的有理同调3-球面族在g = sl 2时的不变量Zr−2*（M，○/），该球面族是通过围绕（2，n）-型环面纽结的整数运算得到的。特别地，我们找到了形式幂级数Z∞（M）∈Q[[h]]在h=q−1中的一个封闭公式，由此可以对所有充分大的素数r导出Zr−2*（M，○/）。我们表明，这种形式的幂级数可以被视为渐近展开，在q=1，一个多值全纯函数的q与1包含在其定义域的边界上。对于这些特殊的流形，其中大多数不是Z-同调球，这扩展了Ohtsuki和Murakami的工作，其中存在有理幂级数。
For any Lie algebra g and integral level k, there is defined an invariant Zk*(M, L) of embeddings of links L in 3‐manifolds M, known as the Witten–Reshetikhin–Turaev invariant. It is known that for links in S3, Zk*(S3, L) is a polynomial in q=exp (2πi/(k+cgv), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2*(M,○/) when g =sl2 for a simple family of rational homology 3‐spheres, obtained by integer surgery around (2, n)‐type torus knots. In particular, we find a closed formula for a formal power series Z∞(M)∈Q[[h]] in h=q−1 from which Zr−2*(M,○/) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z‐homology spheres, this extends work of Ohtsuki and Murakami in which the existence of power series with rational...