Mario O. Bourgoin
We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an application to twisted virtual link theory.
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This paper has been withdrawn by the author due to serious error found in main argument.
Cabling in terms of immersed curves
本研究依托与J. Rasmussen的合作成果及作者此前关于带边Floer同调的工作，推导得出带环面边界流形的Heegaard Floer同调对应的浸入曲线在电缆构造下的行为公式，为纽结Floer同调的研究提供了新的工具。
Immersed Curves and 4-Manifold Invariants
本文研究有界 Heegaard Floer 不变量对应的穿孔环面浸入曲线，通过建立 Fukaya 范畴复合与配边映射态射复合的关系，给出阻碍4维流形及曲面光滑等价的工具。
In 2001, Oestlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion from the circle into the plane to the standard embedding of the circle. We show that this conjecture is false.
Almost $ι$-complexes as immersed curves
本文通过浸入曲线逼近对合 Heegaard Floer 复形，证明了新同态的存在性并重新证明了 $\Theta_{\mathbb{Z}}^3$ 中的 $\mathbb{Z}^{\infty}$ 直和项。
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one half.
Suppose $δ$ is a generic immersed closed curve in the boundary of a 3-manifold M and $δ$ is null-homotopic in M. Then $δ$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.
While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to understand the homotopy type of its subspaces consisting of the curves whose geodesic curvatures are constrained to a prescribed proper open interval. In this article we prove that, under certain circumstances for endpoints and end directions, these subspaces are not homotopically equivalent to the whole space. Moreover, we give an explicit construction of exotic generators for some homotopy and cohomology …
Correction terms of double branched covers and symmetries of immersed curves
该研究借助带边Floer同调的浸入曲线表述，定义新不变量$Δ_{sym}$，得出2分支树状对称链环的双分支覆叠若为L-空间，其自旋$d$-不变量由链环符号差决定的结论。
Existence of Minimal Homotopies for Immersed Planar Curves
本文通过将自交浸入平面曲线提升到高余维空间，利用Douglas的Plateau问题解，发展了一种变分存在性理论，证明了C^1和Lipschitz闭曲线存在面积最小化的同伦。
Deeply Slice Knot Detection via Immersed Curves
本文通过构造一类深切片结，利用Heegaard Floer同调中的不变量和浸入曲线技术，对Kirby列表中Akbulut提出的关于同调3-球面的问题取得进展。
A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this article we describe the Casson invariant of A'Campo's knots as a J^{+/-}-type invariant of the immersed curves. Thus we get an integral generalization of the Gusein-Zade--Natanzon theorem which says that the Arf invariant of a singularity is equal to J^{-}/2(mod 2) of the correspondin…
We characterize those unions of embedded disjoint circles in the 2-sphere which can be the multiple point set of a generic immersion of the 2-sphere into 3-dimensional space in terms of the interlacement of the given circles. Our result is the one higher dimensional analogue of Rosenstiehl's characterization of words being Gauss codes of self-crossing plane curves. Our proof uses a result of Lippner and we further generalize the ideas of Fraysseix and Ossona de Mendez, which leads us to directed interlacement graphs of paired trees and their local complementation.
Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $π_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $π_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques develo…
An endomorphism on immersed curves in the pillowcase
本文研究枕头套中浸入曲线的自同态，通过全纯扰动无迹特征簇函子将标记缠结映射到Weinstein辛部分范畴，并用加倍和八字操作表达，证明其像相同。
Properly immersed curves in arbitrary surfaces via apparent contours on spines of traversing flows
本文通过遍历流脊柱上的表观轮廓研究任意曲面中的正常浸入曲线，将轮廓转化为装饰图以重构曲线，并识别有限局部移动以描述同伦变化。
A curve-based survey of knot Floer homology and concordance invariants
本文综述了纽结Floer同调的浸入曲线解释，并从中提取协变不变量，引入广义$V_s$不变量并给出Upsilon不变量的新曲线描述。
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}…
Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed
该论文证明模空间中全纯曲线在边缘单值为无限阶时是拟等距浸入，并在附加条件下嵌入Teichmüller空间，同时改进了Parshin-Arakelov有限性定理。
The stable cohomology of the moduli space of curves with level structures
本文证明在稳定范围内，带水平结构的曲线模空间与普通曲线模空间的有理上同调相同。
Knot Floer homology as immersed curves
本文将纽结 Floer 同调的数据解释为带标记环面中的带装饰浸入曲线，并通过添加 bounding chain 捕捉 minus 信息，进而给出有理手术的几何计算公式。
We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves gamma: [0,1] -> S^2, i.e., curves with positive geodesic curvature. The space of all such curves with gamma(0) = gamma(1) = e_1 and gamma'(0) = gamma'(1) = e_2 is known to have three connected components X_{-1,c}, X_1, X_{-1}. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X_{-1,c} is contractible, X_1 and X_{-1} are simply connected, pi_2(X_{-1}) contains a copy of Z and pi_2(X_1) contains a copy of Z^2. Also, H^n(X_{1}, R) and H^n(X_{-1}, R) are…
We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g-2. Well known conjectures of Looijenga would imply that this is sharp.
We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.
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