论文标题
虚拟马赛克理论
Virtual Mosaic Knot Theory
论文作者
论文摘要
洛曼科（Lomanoco）和考夫曼（Kauffman）于2008年首次引入了结的镶嵌图，目的是建立量子结系统。从那以后，许多其他人探索了这些结镶嵌图的结构，因为它们本身就是有趣的研究对象。通过加入额外的瓷砖类型来代表虚拟交叉点，加尔图诺（Garduño）已将打结的马赛克概括为虚拟结。但是，对于表面上的结图，虚拟结的另一种解释是激发了这项工作的。通过将经典的马赛克图视为$ 4N $ - 基因和这些多边形的胶合边缘，我们可以在表面上获得结，可以看作是虚拟结。这些虚拟马赛克是我们目前的研究对象。在本文中，我们提供了一组可以在保留结和链接类型的虚拟马赛克上执行的动作，我们表明任何虚拟结或链接都可以表示为虚拟镶嵌物，并且我们提供了与小型古典和虚拟结的虚拟镶嵌数相关的几个计算结果。
Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduño to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as $4n$-gons and gluing edges of these polygons, we obtain knots on surfaces that can be viewed as virtual knots. These virtual mosaics are our present objects of study. In this paper, we provide a set of moves that can be performed on virtual mosaics that preserve knot and link type, we show that any virtual knot or link can be represented as a virtual mosaic, and we provide several computational results related to virtual mosaic numbers for small classical and virtual knots.
