Finite-type invariants for curves on surfaces

Generalized connected sum formula for the Arnold invariants of generic plane curves

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

Based chord diagrams of spherical curves

This paper demonstrates an approach for developing a framework to produce invariants of base-point-free generic spherical curves under some chosen local moves from Reidemeister moves using based chord diagrams. Our invariants not only contain Arnold's classical generic spherical curve invariant but also new invariants.

Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.

Invariant geodesics in the curve complex under point-pushing pseudo-Anosov mapping classes

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

On Finite Order Invariants of Triple Points Free Plane Curves

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as the calculation of knot invariants is based on the study of chord diagrams and connected graphs. E.g., the simplest such invariant is of degree 4 and corresponds to the diagram consisting of two triangles with alternating vertices in a circle in the same way as the simplest knot invariant …

A linking invariant for algebraic curves

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computing this invariant, using a modification of the usual braid monodromy or using the connected numbers introduced by Shirane. As an application, we show that this invariant distinguishes several Zariski pairs, i.e. pairs of curves having same combinatorics, yet different topologies. …

基于曲线的纽结Floer同调与协变不变量综述

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。

Arf invariants of real algebraic curves

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topological machinery not only produces formulae, but it also creates deep connections with the theory of (analytical and topological) multivariable Poincaré series.

平面曲线的多项式不变量

A polynomial invariant of planar curves

本文通过 skein 关系构造平面曲线的多项式不变量，推广了 Arnold 的数值不变量 St。

Effective invariants of braid monodromy and topology of plane curves

In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.

The Galois Action and a Spin Invariant for Prym-Teichmüller Curves in Genus 3

Given a Prym-Teichmüller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the cusps of these Teichmüller curves, extending the results of Bouw and Möller in genus $2$. We use this to show that the components of the genus $3$ Prym-Teichmüller curves are homeomorphic.

J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions

We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the Cauchy-Riemann operator, the curvature of a connection, and the moment map.

Arnold-type Invariants of Curves on Surfaces

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.

Yamada Polynomial and associated link of $θ$-curves

The discovery of polynomial invariants of knots and links, ignited by V. F. R. Jones, leads to the formulation of polynomial invariants of spatial graphs. The Yamada polynomial, one of such invariants, is frequently utilized for practical distinguishment of spatial graphs. Especially for $θ$-curves, the polynomial is an ambient isotopy invariant after a normalization. On the other hand, to each $θ$-curve, a 3-component link can be associated as an ambient isotopy invariant. The benefit of associated links is that invariants of links can be utilized as invariants of $θ$-curves. In this paper we…

Fundamental groups of complements of plane curves and symplectic invariants

Introducing the notion of stabilized fundamental group for the complement of a branch curve in $CP^2$, we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.