Ying Gu, Xuezhi Zhao
Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
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We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams."
Given a compact orientable surface $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential simple loops on $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in $\bold R^{\Cal S(Σ)}$ and $P(\bold R^{\Cal S(Σ)})$ defining Thurston's space of measured laminations and Thurston's compactification of the Teichmüller space. These equations are not only piecewise integral linear but also semi-real algebraic.
Given two free homotopy classes $α_1, α_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(α_1, α_2)$ of loops in these two classes. We show that for $α_1\neqα_2$ the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of $α_1$ and $α_2$ is equal to $m(α_1, α_2)$. Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of $α_1$ and $α_2$. The main result of this paper in the case where $α_1, α_2$ do not contain different powers of the same loop first appeared in …
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric intersection number of the curves is greater than zero when their algebraic intersection number is zero. The value, computed from two elements of the fundamental group of the surface corresponding to the curves, is found in the difference between one of the elements and its image of the action of Dehn twist along the other. To give a description of the difference sy…
We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.
We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem of detecting the geometric intersection between separating simple closed curves in a representation.
We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics
We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorphisms of PML(S) that preserve the zero sets of these intersection functions.
In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold $X$ with second positive Betti number $b_2^+(X)=1$. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this…
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algorithms to (1) compute the geometric intersection number of $c$ in $O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+\ell^4)$ time, (3) decide if the geometric intersection number of $c$ is zero, i.e. if $c$ is hom…
We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the surfaces by certain graphs.
For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\bar{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb R$-trees. The projectivization of $\bar{cv}(F_N)$ provi…
We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the bou…
Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[π]$ of the fundamental group $π$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra $\mathbb{Q}[π]$ is completed with respect to powers of its augmentation ideal and is app…
In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings. We show that $μ$ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev's virtual string cobracket. We use Turaev's based matrices to describe strings $α$ such that $μ$ gives a formula fo…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the curve graphs and the mapping class groups.
Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). Let S^3_1 and S^3_2 intersect transversely. Then the intersection C of S^3_1 and S^3_2 is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), and a pair of 3-knots, S^3_i in S^5 (i=1,2). Conversely let (L_1,L_2) be a pair of 1-links and (X_1,X_2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L_1,L_2) is obtained as the intersection of the 3-knots X_1 and X_2 as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant…
Commensurability and quasi-isometry classification for one vertex one loop tubular groups
本文对单顶点单环管状群按边映射交数分类：非零交数时全体可公度且拟等距；零交数时有两个拟等距类和无穷可公度类，两类间不拟等距。
We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson homomorphism.
Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichmüll…
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…
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