Homotopy on spatial graphs and the Sato-Levine invariant

Link-homotopy classes of 4-component links and claspers

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of 4-component links were classified by Levine with enormous algebraic computations. We modify the results by using Habiro's clasper theory. The new classification gives more symmetrical and schematic points of view to the link-homotopy classes of 4-component links. As applications, we give several new subsets of the link-homotopy classes of 4-component links which are classified by comparable invariants and give an algorithm which determi…

Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $μ$ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the so-called Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.

Stable concordance of knots in 3-manifolds

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…

关于链同伦意义下的辫与链环

On braids and links up to link-homotopy

该研究基于Habiro的clasper同伦演算，一方面定义计算同伦辫群的忠实线性表示，另一方面给出4分量链环链同伦分类的几何证明并推进代数分裂下5分量链环的分类。

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructi…

Garoufalidis-Levine 关于从 3-流形到 3-环面的 $\mathbb{Z}π$-同调等价的有限型不变量

Garoufalidis-Levine's finite type invariants for $\mathbb{Z}π$-homology equivalences from 3-manifolds to the 3-torus

本文证明 Garoufalidis-Levine 对到 3-环面的 $\mathbb{Z}π$-同调等价的有限型不变量满射在有理数上为同构，并利用 Fukaya Morse 同调论构造微扰不变量，推广 Casson 不变量。

Surgery on links of linking number zero and the Heegaard Floer $d$-invariant

We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $ν^…

The Milnor invariants of clover links

J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most $2k+1$ are well-defined if those of length at most $k$ vanish, and that the Milnor numbers of length at least $2k+2$ are not well-defined if those of length $k+1$ survive. For a clover link $c$ with the Milnor numbers of length at most $k$ vanishing, we show that the Milnor number $μ_c(I)$ for a sequence $I$ is well-defined up to the greatest common devisor of $μ_{c}(J)'s$, where $J$ is a subsequence of $I$ obtained b…

Clasper 一致性、Whitney 塔与重复 Milnor 不变量

Clasper Concordance, Whitney towers and repeating Milnor invariants

本文证明链环在 4-球中界定度为 k 的 Whitney 塔等价于与平凡链环 $C_k$-一致，并利用重复扭曲 Whitney 塔分类了扭曲自 $C_k$-一致性，基于 Milnor 和 Arf 不变量。

n-Quasi-isotopy: I. Questions of nilpotence

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k=1,2,...; all of the…

n-quasi-isotopy: III. Engel conditions

In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant β^k, provided k>2, and a series of \barμ-invariants, starting with \barμ(111112122) for k=3, also fall in this gap. In fact, all \barμ-invariants where each index occurs at most k+1 times, except perhaps for one occuring k+2 times, can be extracted from G_k, and if they vanish, G…

焊接图、Wirtinger群与纽结穿孔球面

Welded graphs, Wirtinger groups and knotted punctured spheres

本文建立焊接图的图示理论并扩展 Satoh 的 Tube 映射，用以分类纽结穿孔球面、刻画其 Wirtinger 群，并提出保持 Tube 映射不变的 Υ 局部移动。

Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic spin TQFTs are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gaugi…