论文标题
结的强度和弱（1，2，3）
Strong and weak (1, 2, 3) homotopies on knot projections
论文作者
论文摘要
打结投影是从圆中浸入二维球体的一般浸入的图像。我们可以通过局部替换三种类型的打结投影（称为Reidemeister Moves）在任何两个结的预测之间找到同构。本文定义了一个称为弱（1、2、3）同型的结次的相等关系，该相关性由1型，弱2型和弱3型的reidemister移动组成。本文定义了第一个在弱（1、2、3）同型弱（1、2、3）下的非平凡不变的。我们使用这种不变性来表明，有无限数量的弱（1、2、3）均匀等效类别的打结投影类别。相比之下，所有等效类别的打结投影类别由三型的其他变体组成，即（1，强2型，强3型，强3）（1，弱2型，强型3）和（1，强2型，强2，弱2，弱3）的移动是合同的。
A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first non-trivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e., Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.
