Twisted Alexander polynomial and Reidemeister torsion
Twisted Alexander polynomial and Reidemeister torsion
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扭曲亚历山大多项式和 Reidemeister 挠率
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发表时间：
1996
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通讯作者：
Teruaki Kitano
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作者：
Teruaki Kitano
1992年，Wada[4]定义了有限可呈现群的扭曲亚历山大多项式。让Γ成为一个有限的体面组。我们假设阿贝尔化Γ/[Γ， Γ]是一个自由阿贝尔群Tr - (t1)。， tr ttj = tjti)的秩为r。然后我们将分配一个洛朗多项式Δr) P (t1，…， tr)具有唯一的分解域i？-每个线性表示p的系数：Γ -> GL（n R）。我们称其为与p相关的Γ的扭曲Alexander多项式。为简单起见，我们假设R是实数域R， p的像包含在SL（n] R）中。因为我们主要对结群的情况感兴趣，因此我们假设Γ是一个结群。设kcs是一个结点，E是K的外部结点，我们用表示Γ的正则化
In 1992, Wada [4] defined the twisted Alexander polynomial for finitely presentable groups. Let Γ be a finitely presentable group. We suppose that the abelianization Γ/[Γ, Γ] is a free abelian group Tr — ( t l 5 . . . , tr ttj = tjti) of rank r. Then we will assign a Laurent polynomial Δr ) P (t i , . . . , tr) with a unique factorization domain i?-coefficients to each linear representation p : Γ -> GL(n R). We call it the twisted Alexander polynomial of Γ associated to p. For simplicity, we suppose that R is the real number field R and the image of p is included in SL(n] R). Because we are mainly interested in the case of the group of a knot, hereafter we suppose that Γ is a knot group. Let K C S be a knot and E its exterior of K. We denote the canonical abelianization of Γ by