arXiv math/02110572002-11-04math.GT

A surgery formula for the 2-loop piece of the LMO invariant of a pair

Andrew Kricker

备注：: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper11.abs.html
发表信息：: Geom. Topol. Monogr. 4 (2002) 161-181

英文摘要：

Let Θ(M,K) denote the 2-loop piece of (the logarithm of) the LMO invariant of a knot K in M, a ZHS^3. Forgetting the knot (by which we mean setting diagrams with legs to zero) specialises Θ(M,K) to λ(M), Casson's invariant. This note describes an extension of Casson's surgery formula for his invariant to Θ(M,K). To be precise, we describe the effect on Θ(M,K) of a surgery on a knot which together with K forms a boundary link in M. Whilst the presented formula does not characterise Θ(M,K), it does allow some insight into the underlying topology.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in S^{3}

We use the LMO invariant to find constraints for a knot to admit a purely or reflectively cosmetic surgery. We also get a constraint for knots to admit a Lens space surgery, and some information for characterizing slopes.

2017-04-24math.GT

A Rational Surgery Formula for the LMO Invariant

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^3. Our main tool is a careful use of the Aarhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered space…

2000-07-07math.GTmath.QA

A splicing formula for the LMO invariant

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's fo…

2020-08-15math.GTmath.QA

LMO谱：分解同调与雅可比图代数的E₃结构

The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra

本研究利用分解同调定义三维流形LMO不变量的范畴化版本LMO谱，证明雅可比图代数的同伦E₃-代数结构，推导通用手术公式，并构造H₁-装饰LMO不变量以区分经典LMO不变量无法辨别的一对透镜空间。

2025-10-06math-phmath.ATmath.GT

The Structure and Enumeration of Link Projections

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.

1994-11-07math.GT

A note on chirally cosmetic surgery on cable knots

We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery as long as the JSJ piece of knot exterior does not contain $(2,r)$-torus exterior. We also show that an iterated torus knot other than $(2,p)$-torus knot does not admit chirally cosmetic surgery.

2019-05-16math.GT

更多相关论文收起更多论文

闭3-流形的Le-Murakami-Ohtsuki不变量综述

A survey on the Le-Murakami-Ohtsuki invariant for closed 3-manifolds

本文综述闭3-流形LMO不变量的原始构造方法，引入Kirby结构、pre-LMO结构等组合工具，结合Kontsevich积分得到两族乘法不变量，阐述LMO不变量的导出过程及相关唯一性结果，并补充了未被文献明确讨论的一族不变量的性质。

2024-10-30math.GTmath.QA

The lines of the Kontsevich integral and Rozansky's rationality conjecture

This work develops some technology for accessing the loop expansion of the Kontsevich integral of a knot. The setting is an application of the LMO invariant to certain surgery presentations of knots by framed links in the solid torus. A consequence of this technology is a certain recent conjecture of Rozansky's. Rozansky conjectured that the Kontsevich integral could be organised into a series of ``lines'' which could be represented by finite $\Qset$-linear combinations of diagrams whose edges were labelled, in an appropriate sense, with rational functions. Furthermore, the conjecture requires…

2000-05-31math.GTmath.QA

通用不变量、康威多项式和卡森-沃克-莱斯科普不变量

Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant

Univ. Grenoble Alpes, CNRS, IF（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，IF）

本文通过图示和组合方法给出了闭3流形卡森-沃克-莱斯科普不变量的手术公式，揭示了康威多项式系数与 Kontsevich 积分系数的关系。

2024-07-22math.GT

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterize…

2023-01-15math.DGmath.GT

Computing a Link Diagram from its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the fi…

2023-03-16cs.CGmath.GT