相关论文: On the Wu invariants for immersions of a graph int…
Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…
A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu…
We introduce invariants of spatial graphs related to the Wu invariant and the Simon invariant, and apply them to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of embedded…
Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies.…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^3, with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T \oplus…
Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We…
In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R^{2n-1} to locally generic mappings. The main result is that if n=2 then two mappings with singularities are regularly homotopic…
Our aim is to determine the regular homotopy classes of immersions related to Arnol'd's simple singularities. For every type of simple singularities, we determine the regular homotopy class of the inclusion map of the link into the…
In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…
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…
A drawing in the plane ($\mathbb{R}^2$) of a graph $G=(V,E)$ equipped with a function $\gamma: V \rightarrow \mathbb{N}$ is \emph{$x$-bounded} if (i) $x(u) <x(v)$ whenever $\gamma(u)<\gamma(v)$ and (ii) $\gamma(u)\leq\gamma(w)\leq…
Graph maps that are homotopic to the identity and that permute the vertices are studied. Given a periodic point for such a map, a {\em rotation element} is defined in terms of the fundamental group. A number of results are proved about the…
We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where…
We give a necessary condition for two diagrams of $3$-regular spatial graphs with the same underlying abstract graph $G$ to represent isotopic spatial graphs. The test works by reading off the writhes of the knot diagrams coming from a…
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map…
We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we…
An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…