By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.
The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a combinatorial redefinition of these two operations through a natural identification of the homotopy classes of closed curves on the surface with the cyclic words in the generators and their inverses of the fundamental group of the surface. We present a new approach to give a combinatorial definition of the bracket and cobracket, focusing on the information given …
In this paper, we introduce two functions such that the subtraction corresponds to the Milnor's triple linking number; the addition obtains a new integer-valued link homotopy invariant of $3$-component links. We also have found a series of integer-valued invariants derived from four terms whose sum equals the Milnor's triple linking number. We apply this structure to give invariants of $3$-bouquet graphs.
We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin suspension, and both were unexpected for the author, because the only known direct ways to extract $μ$ and $β$ from invariants of maps between configuration spaces involved some homotopy theory (Whitehead products and the stable Hopf invariant, respectively).
Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2
我们描述了一种新的方法，三重连接不变量和积分，旨在更简单，更广泛和更自然的适用性，以寻找高阶螺旋的流体流动和磁场。对于欧氏空间中的每一个三分量链，我们将一个从3-环面到2-球面的几何上自然的广义Gauss映射联系起来，并证明了将链分类到链同伦的成对环绕数和Milnor三元环绕数对应于将其广义Gauss映射分类到同伦的Pontryagin不变量.这可以被看作是一个熟悉事实的自然扩张，即三维空间中一个二元链的环绕数是它从2-环面到2-球面的高斯映射的次数。当两两环绕数均为零时，我们给出了一个类似于两两环绕数的Gauss积分的三重环数的积分公式，但仿照J.H.C. Whitehead's integral formula for the Hopf invariant.该公式中的被积函数在三维空间中保持方向不变的刚体运动下是几何上自然的，而积分本身可以看作是三维环面上相关向量场的螺旋度。在本系列的第一篇论文[math.GT 1101.3374]中，我们对3-球面中的三分量连杆进行了此操作。Komendarczyk应用这种方法在特殊情况下，推导出一个高阶螺旋度的磁场，其普通螺旋度为零，并获得从这个非零的下界的字段能量。
