Kontsevich积分的环展开、零点移动和S-等价

On the LMO Invariant, the Wheeling Theorem, and the Aarhus Integral

In a previous paper, we generalized the definition of the framed Kontsevich integral initially presented by Le and Murakami. We also defined an isotopy invariant $\widetilde{Z}_f$ that is well-behaved under band sum moves. Using this invariant we study the construction of the LMO invariant, the Wheeling Theorem, and the Aarhus integral.

Knot invariants derived from the equivariant linking pairing

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M;Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q' of null-homologous knots in rational homology …

Rational Blanchfield forms, S-equivalence, and null LP-surgeries

Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replacements of null-homologous rational homology handlebodies of M\K by other such handlebodies with identical Lagrangian. A null Lagrangian-preserving surgery induces a canonical isomorphism between the rational Alexander modules of the involved pairs, which preserves the Blanchfield …

边界环的一个有理非交换不变量

A rational noncommutative invariant of boundary links

1999年，Rozansky证明了一个纽结的Kontsevich积分的合理表示的存在。粗略地说，这种合理的介绍Kontsevich积分将和正式的幂级数到有理函数与规定的消除。罗赞斯基的猜想很快被第二作者证明。我们开始我们的文件，审查Rozansky的猜想和主要思想，导致其证明。将这个猜想扩展到链接的自然问题导致了边界链接类，并在这种情况下证明了Rozansky猜想。一个微妙的问题是，用头发的指数代替珠子的“头发”贴图不是1-1。这就提出了一个问题，是否存在一个有理不变式的边界链接在一个适当的空间中的三价图的边缘装饰的有理函数在noncommuting变量。本文的一个主要结果是构建这样一个不变量，使用所谓的手术视图的边界链接和开发后，一个正式的图解高斯积分。由于我们的不变量是Kontsevich积分的许多有理形式之一，人们可能会问我们的不变量在某种意义上是否是规范的。我们证明，这确实是这样的情况下，通过公理化的特点，我们的不变量作为一个普遍的有限型不变的边界链接相对于零移动。最后讨论了我们的有理不变式与同调运算的关系，并给出了它在低维拓扑中的一些应用。

Kirby calculus for null-homologous framed links in 3-manifolds

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed links in a 3-manifold. This result involves a new kind of moves, called IHX-moves, which are closely related to the IHX relation in the theory of finite type invariants. When the first homology group of M is free abelian, we give a refinement of this result to \pm1-framed, algebra…

基于带通和双Delta移动的有限型不变量

Finite type invariants based on the band-pass and doubled delta moves

我们研究了有限型结不变量的推广，通过替换交叉变化的Vassiliev绞关系由其他一些本地移动，详细分析了带通和双三角洲移动。使用辫子理论的技术，我们表明，对于一个大的类的本地移动，广义Goussarov的n-等价类的结形成组下连接和。（谷山（Taniyama）和安原（Yasuhara）以前也得到过类似的结果，但方法不同。）结果表明，原始带通有限型不变量本质上与标准原始有限型不变量一致，但对于双δ移动，情况更有趣。完全0度双δ不变量是纽结的$S$ -等价类。在此背景下，我们推广了Murakami和Ohtsuki的结果，证明了在没有2-挠的阿贝尔群中取值的S-等价的唯一原始Vassiliev不变量来自Alexander-Conway多项式。我们开始分析度1双δ不变量，考虑哪些瓦西里耶夫不变量是双δ度1，发现在每个奇瓦西里耶夫度中正好有一个这样的不变量，在每个偶瓦西里耶夫度中最多有一个（Z/2值）。分析更高的双δ度，我们观察到欧拉度n+1。Kontsevich积分的Garoufaldet和Kricker有理提升的一部分是双δ度$2n$不变量。

On Kirby calculus for null-homotopic framed links in 3-manifolds

Kirby proved that two framed links in S^3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, oriented 3-manifold to be related by a finite sequence of these moves. The purpose of this paper is twofold. We first give a generalization of Fenn and Rourke's result to 3-manifolds with boundary. Then we apply this result to the case of framed links whose components are null-homotopi…

On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M)/Torsion. This article is devoted to the study of the invariant Q. We prove many properties for this invariant including two surgery formulae. Via surgery, the invariant Q is equivalent to an invariant of null-homologous kno…

Morse structures on open books

We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $\mathbb{R}^3$ to arbitrary closed contact $3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous L…