arXiv research

Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

Novel approach for large genus intersection number asymptotics.

The paper finds diffeomorphic complex intersections with distinct Hodge numbers.

In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…

The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.

The study limits how many parts regular simplicial partitions can overlap.

Finite intersection numbers between horizontal foliations of quadratic differentials.

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 $S^2\times S^1$.

The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.

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.

Improved bounds on geodesic intersections on hyperbolic surfaces.

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.

New polynomials defined for virtual knots, calculated up to crossing 4.

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…

Conditions for curves on a torus with specific pairwise intersections.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…

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…

Sharp lower bound on fold singularities self-intersections.

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.

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.

We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the …

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

The study finds an upper limit for the number of minimal origami pairs on a surface.

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 algo…

The paper proves the exact number of singular points in the intersection of convex shapes.

We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.

The paper constructs minimal coherent filling pairs on surfaces.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free…

Study intersection cohomology and Lagrangian fibrations in symplectic varieties.

Study on combinatorial $k$-systoles on surfaces, showing growth in intersection numbers.

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given…

The study counts geodesics on curved surfaces with specific intersections.

The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of matrices of positive corank is an example of such a set. It turns out that the inte…

Suppose a smooth planar curve $γ$ is $2π$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $γ$ self-intersects, then it has a segment of length $\ell- 2π$ on which it self-intersects and somewhere its curvature is at least $2π/(\ell - 2π)$. The proof involves the projection $Γ$ …

Defines quantum intersection number on pants decompositions and relates it to hyperbolic geometry.

The paper proves uniform stable radius and Milnor number equality for specific mappings.

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…

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 que…

The study counts curves on a once-punctured torus with self-intersections.

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Determined the balanced cone of a specific geometric space.

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…

New origamis found for surfaces with minimal intersections.