A free-group valued invariant of free knots

On long knots in the full torus

The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian knots, knotoids, 3-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [6]. Long knots naturally appear in the study of classical knots [1, 8].

标架旋转纽结和曲面链环的圆值莫尔斯理论

Circle-valued Morse theory for frame spun knots and surface-links

设N是（k+2）维球面的k维闭定向子流形;用C（N）表示它的补流形。用x表示C（N）中与N对偶的1维上同调类。C（N）的Morse-Novikov数定义为从C（N）到圆的正则Morse映射f的临界点的最小可能个数，使得f属于x。本文的第一部分研究了当N是与m-纽结K相关联的扭标架自旋纽结时的情形。我们得到了N和K的Morse-Novikov数之间的关系式，推广了D.罗斯曼和E.C.塞曼关于纺结的纤维化。在第二部分中，我们将所得结果应用于4-球面中表面链的Morse-Novikov数的计算。

Homology of ternary algebras yielding invariants of knots and knotted surfaces

We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of Reidemeister, Yoshikawa, and Roseman moves. Our homology has a degenerate subcomplex. The normalized homology yields invariants of knots and knotted surfaces.

Nielsen数是一个纽结不变量

Nielsen number is a knot invariant

通过表示簇证明了映射的Nielsen数是纽结不变量

同调纤维纽结高阶Alexander不变量的因式分解公式及计算

Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots

同调纤维纽结是其外部满足与纤维纽结相同的同调条件的纽结。在我们以前的论文中，我们观察到，对于这样的结，高阶亚历山大不变量定义的Cochran，Harvey和Friedl一般分解成部分的Magnus矩阵和一定的Reidemeister挠，这两个都被称为在一个表面上的同调圆柱的不变量。本文将不变量限定在与其纽结群的亚交换群有关的情形下，对不变量作了更详细的研究，并给出了具体的计算。我们提供的例子明确计算的不变量为所有12个交叉非纤维同调纤维结。

最小辫子：结和链的完全不变量

Minimum Braids: A Complete Invariant of Knots and Links

最小辫子是结和链的完全不变量。本文定义了最小辫子，介绍了如何生成它们，提出了表结多达10个交叉点和定向链接多达9个交叉点，并使用最小辫子研究图树，双射性，unknotting号码，和周期表。