Twisted Alexander Polynomials of Hyperbolic Knots
Twisted Alexander Polynomials of Hyperbolic Knots
复制标题
双曲结的扭曲亚历山大多项式
DOI：
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发表时间：
2011
影响因子：
0.5
通讯作者：
Nicholas Jackson
中科院分区：
文献类型：
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作者：
N. Dunfield;Stefan Friedl;Nicholas Jackson
通过将完整表示提升到，我们研究了与整数同调3-球面上的双曲纽结自然相关的扭曲Alexander多项式。它是一个明确的对称洛朗多项式，其系数位于纽结的迹域内。它包含有关属、纤毛和手性的信息，而且功能强大，有时足以检测到突变。我们数值计算了S 3中最多有15个交叉点的所有313209个双曲纽结的这个不变量，发现在所有情况下，它都给出了纽结的亏格的一个尖锐的界限，并确定了纤化和手性。我们还研究了这种扭曲的Alexander多项式在纽结群的特征标簇的不可约分量X0中移动时如何变化。我们展示了如何通过一个多项式一次理解所有这些多项式，该多项式的系数位于X0的函数域。我们用它来帮助解释在S 3中观察到的纽结的一些模式，并探索这个普适多项式和与理想点有关的曲面的卡勒-沙伦理论之间的潜在关系。
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to . It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the -character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0. We use this to help explain some of the patterns observed for knots in S 3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.