An integer valued SU(3) Casson invariant

Ribbon-moves of 2-knots: the Farber-Levine pairing and the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman $\widetildeη$-invariants of 2-knots

Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of $K$ is isomorphic to that of K' as Z[Z] modules. Let K be a 2-knot which is ribbon-move equivalent to the trivial knot. Then the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman Q/Z-valued \widetildeη-invariants of K for Z_d is zero. (d is a natural number. d>2.).

非平凡容许 $U(2)$ 丛的 Klein 四群联络与 Casson 不变量

Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles

本文研究非平凡容许 $U(2)$ 丛的 Casson 不变量，通过模2上同调环及 Klein 四群完整群平坦联络的计数公式，刻画其偶性来源。

Floer homologies for Lagrangian intersections and instantons

In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued invariant for any oriented integral homology 3-sphere Y via beautiful constructions on representation spaces (see [1] for an exposition). The Casson invariant λ(Y) is roughly defined by measuring the oriented number of irreducible representations of the fundamental group π_1(Y) in SU(2). Such an invariant generalized the Rohlin invariant and gives surprising corollaries in low dimensional topology.

Homological Casson type invariant of knotoids

We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing smal…

4维流形同胚的卡森-沙利文不变量

The Casson-Sullivan invariant for homeomorphisms of 4-manifolds

研究光滑4维流形同胚的卡森-沙利文不变量，证明用单个\(S^2\times S^2\)稳定化后可完全实现，给出拓扑同痕曲面在充分外部稳定化后变为光滑同痕等应用，还展示不稳定化情形及相关例子，可用于找特定光滑结构。

SL(2)实形式的统一Casson-Lin不变量

A unified Casson-Lin invariant for the real forms of SL(2)

本文提出了一种统一的Casson-Lin不变量，用于研究SL(2,R)表示与SU(2)表示的关系，并证明了其通过拓扑假设可推导出多种表示的存在性。

Finite type invariants of cyclic branched covers

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its sign…

The coefficients of the Jones polynomial

It has been known that, the coefficients of the series expansion of the Jones polynomial evaluated at $e^x$ are rational valued Vassiliev invariants . In this article, we calculate minimal multiplying factor, λ, needed for these rational valued invariants to become integer valued Vassiliev invariants. By doing that we obtain a set of integer-valued Vassiliev invariants.

An integer valued bi-invariant metric on the group of contactomorphisms of R^2n x S^1

In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity. We also prove that the contactomorphism group is unbounded with respect to this metric.

A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori

Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold $X$ with the integral homology of $S^{1}\times S^{3}$, and an embedded torus $T$ inside $X$ such that $H_{1}(T;\mathbb{Z})$ surjects onto $H_{1}(X;\…