论文标题
通过跨越表面的虚拟结的新不变性
New invariants for virtual knots via spanning surfaces
论文作者
论文摘要
我们定义了三种不同类型的跨度表面，以在增厚的表面中打结。我们使用这些引入新的Seifert矩阵，亚历山大型多项式，属和签名不变。这些亚历山大多项式延伸到虚拟结，可能会阻碍虚拟结的经典。此外，它可以区分增厚表面的结与镜子到同位素的结。我们还提出了一些Heegaard浮子同源性的几种结构，以使其在增厚的表面中打结，并举例说明为什么它们不是稳定不变的。但是，我们可以通过最少的属代表来定义虚拟结的浮点同源性。最后，我们使用behrens-golla $δ$ invariant来阻碍结的稳定。
We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens-Golla $δ$-invariant to obstruct a knot from being a stabilization of another.
