arXiv research

Sharp lower bound on fold singularities self-intersections.

Study mapping class group orbits of curves with self-intersections on surfaces.

The paper counts orbits of curves with up to K self-intersections on a surface.

The paper describes algebraic operations on surface fundamental groups.

We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h-2 is the only universal bound on the self-intersection number of a section of any such genus g bun…

Survey on algebraic aspects of Turaev cobracket and related problems.

It is shown that Schneiderman and Teichner's invariant $τ$ detects the example due to Kirk of a link map $f:S^2_+\cup S^2_-\to S^4$ for which the restricted map $f|S^2_+:S^2_+\to S^4-f(S^2_-)$ has vanishing Wall self-intersection but is not homotopic to an embedding.

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

Extended Birman-Series result to geodesics with infinitely many self-intersections.

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

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.

We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…

Study on shortest geodesics with self-intersections on hyperbolic surfaces.

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

Time-delayed embeddings avoid self-intersections for high enough delay.

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence…

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

Upper and lower bounds for curve types on surfaces found.

The study examines elastic curves with self-intersections and their properties.

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

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 $Γ$ …

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

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…

The study finds that most minimal surfaces in generic 4D manifolds intersect in complex ways.

Generic potential primes have no self-intersections or intersections.

Study on parity of singular set components of maps to surfaces.

We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question …

Geodesics with bounded angles have zero Hausdorff dimension.

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…

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 …

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

Given an orientable surface with boundary and a free homotopy class, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self intersection.

The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.

Solves Plateau's problem for curves with self-intersections.

Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Ge…

Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.

The paper proves existence of minimal homotopies for immersed planar curves.

We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximab…

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…

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…

The paper proves an adjunction inequality for Real embedded surfaces in 4-manifolds.

Study of Hamiltonian flows on character varieties for self-intersecting curves.

Minimal geodesics on hyperbolic surfaces are long.

2D complexes can be almost-embedded in 4D space without self-intersections.

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L…

The paper constructs hyperbolic metrics for essential curves with self-intersections and lifts them simply.

The study proves geodesic loops and chords without intersections for specific metrics.

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…