最小自相交表示可视化的组合算法

双曲面上的最短k-测地线

Shortest k-Geodesics on Hyperbolic Surfaces

本文研究双曲面上闭测地线的长度与自交数的关系，改进了现有估计，将最大自交数的渐进行界从512降至128。

Product and coproduct in string topology

Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We prove the following new geometric property of the dual homology coproduct: the nonvanishing of the k-th iterate of the coproduct on a homology class ensures the existence of a loop with a (k+1)-fold self-intersection in every representative of the class. For spheres and projective…

$\mathbb{R}^4$中极小曲面的奇异结点

Singularity Knots of Minimal Surfaces in $\mathbb{R}^4$

我们研究了$\mathbb{S}^3$中的节点，该节点是由$\mathbb{R}^4$中的极小曲面与以分支点为中心的小3-球面相交而得到的。我们构造新的最小结的例子。特别是，我们证明了非纤维最小结的存在。我们表明，简单的最小的结是可逆的或完全的amphicheiral的，这会产生一个障碍，一个给定的结是一个迭代结的极小曲面。这些结的性质和不变量，如代数交叉数的辫子代表和亚历山大多项式进行了研究。

An algebraic characterization of simple closed curves on surfaces with boundary

We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permut…

Counting Mapping Class group orbits on hyperbolic surfaces

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[γ]$ of a closed curve $γ\in π_{1}(S_{g,n})$ contains a unique closed geodesic on $X$. Let $\ell_γ(X)$ denote the hyperbolic length of the geodesic representative of $γ$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length $\leq L$ on $X$ with at mo…

k-曲线图的自同构

Automorphisms of the k-curve graph

本文研究有限型可定向曲面上的 k-曲线图，证明当 k 相对欧拉示性数足够小时其自同构群同构于扩展映射类群，并将结论推广至 systolic complex，解决 Schmutz Schaller 的猜想。