Virtual knot groups and almost classical knots
Virtual knot groups and almost classical knots
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DOI：
10.4064/fm80-9-2016
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发表时间：
2015-06
期刊：
影响因子：
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通讯作者：
H. Boden;Robin Gaudreau;Eric Harper;Andrew J. Nicas;Lindsay White
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文献类型：
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作者：
H. Boden;Robin Gaudreau;Eric Harper;Andrew J. Nicas;Lindsay White
我们定义了虚拟结的群值不变量，并将其与虚拟结的其他各种群值不变量相关联，包括 Silver-Williams 的扩展群以及 Manturov 和 Bardakov-Bellingeri 的 qudle 群。如果虚拟结允许具有亚历山大编号的图，则该虚拟结被称为几乎经典的，在这种情况下，我们将群因子证明为通常的结群和 Z 的自由乘积。我们为 mod p 几乎经典的结建立了一个类似的公式，并且我们使用这些结果来推导虚拟结 K 成为 mod p 几乎经典的障碍。被视为加厚表面中的结，几乎经典结对应于同调平凡的结。我们证明他们承认 Seifert 曲面并将其亚历山大不变量与相关无限循环覆盖的同源性联系起来。我们证明了第一个亚历山大理想是主要的，恢复了中村等人首先证明的结果。使用不同的方法。结果表明，亚历山大多项式满足绞纱关系，其次数给出了 Seifert 属的下界。我们将最多 6 个交叉点的几乎经典结制成表格，并确定它们的亚历山大多项式和虚拟属。
We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to 6 crossings and determine their Alexander polynomials and virtual genus.