Asymptotics of the colored Jones function of a knot
Asymptotics of the colored Jones function of a knot
复制标题
结的彩色琼斯函数的渐近
DOI：
10.2140/gt.2011.15.2135
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发表时间：
2005
影响因子：
2
通讯作者：
Thang T. Q. Lê
中科院分区：
文献类型：
--
作者：
S. Garoufalidis;Thang T. Q. Lê
对于三维空间中的一个结，我们可以将一个洛朗多项式序列联系起来，它的第n项是第n个有色琼斯多项式。研究了当固定复数且n趋于无穷时，第n个有色琼斯多项式的值在e /n处的渐近性质。当是一个足够小的复数时，我们分析了这种对1/n的所有阶的渐近行为。此外，我们给出了第10个有色琼斯多项式的系数和次的上界，并将其应用于广义体积猜想的上界。Agol-Dunfield-Storm-W的作品瑟斯顿暗示我们的界是渐近最优的。此外，我们还给出了广义体积猜想在2 i附近的结果。我们的证明关键地使用了哈比罗的有色琼斯函数的分圈展开。
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the nth colored Jones polynomial at e�/n, whenis a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1/n whenis a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of thenth colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when � is near 2�i. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.