The homological arrow polynomial for virtual links
The homological arrow polynomial for virtual links
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虚拟链接的同调箭头多项式
DOI：
10.1142/s0218216523500050
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发表时间：
2022
影响因子：
0.5
通讯作者：
Kyle A. Miller
中科院分区：
文献类型：
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作者：
Kyle A. Miller
箭头多项式是框架定向虚连杆的不变量，它推广了虚考夫曼括号。本文定义了同调箭头多项式，将箭头多项式推广到带标号构件的框架定向虚链路。关键的观察是，给定一个链接在一个加厚的表面，同源类的链接定义一个功能的表面的绞链模块，并通过将其应用到图像的链接在绞链模块这给出了一个虚拟的链接不变。我们给出了一个图形演算的同调箭头多项式采取通常的图考夫曼括号，并包括标记的“须“，记录交叉数与每个标记的组件的链接。利用同调箭头多项式研究了$（\mathbb{Z}/n\mathbb{Z}）$-空同调虚链路和棋盘可染性，给出了一种新的方法来完成Imabeppu关于至多四个交叉点的虚链路棋盘可染性的刻画.我们还证明了Kauffman-Murasugi-Thistlethwaite定理的一个版本，即“h-约化“图$D$的同调箭头多项式的求值宽度为$4（c（D）-g（D）+1）$。
The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface's skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled"whiskers"that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study $(\mathbb{Z}/n\mathbb{Z})$-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu's characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman-Murasugi-Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an"h-reduced"diagram $D$ is $4(c(D)-g(D)+1)$.
DOI：
10.4064/fm80-9-2016
发表时间：
2015-06
期刊：
arXiv: Geometric Topology
影响因子：
--
作者：
H. Boden;Robin Gaudreau;Eric Harper;Andrew J. Nicas;Lindsay White
通讯作者：
H. Boden;Robin Gaudreau;Eric Harper;Andrew J. Nicas;Lindsay White