论文标题
关于琼斯多项式的概括及其对legendrian结的分类
On a generalization of Jones polynomial and its categorification for Legendrian Knots
论文作者
论文摘要
在本文中，我们探索了传统结的多项式不变，这是琼斯多项式（拓扑）结的自然扩展。为此，为Legendrian打结的前突出引入了一种新型的绞线关系。此外，我们为Legendrian结提供了多项式不变的分类，这是Khovanov的自然扩展。 Legendrian结的Thurston-Bennequin不变型在同源性的构造中自然而然地作为换档。多项式不变及其分类的构造是自然的，因为如果我们将legendrian结视为结（即，我们忘记结上的几何形状），那么我们分别恢复了琼斯多项式和Khovanov同源性。最后，我们讨论了这些不变的人的优势和局限性。
In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology for knots. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade-shift. The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.
