Plane curves in an immersed graph in $R^2$

@article{Sakamoto2012PlaneCI,
  title={Plane curves in an immersed graph in \$R^2\$},
  author={Marisa Sakamoto and Kouki Taniyama},
  journal={arXiv: Geometric Topology},
  year={2012}
}
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 sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one half. 
Sub-chord diagrams of knot projections
A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve provides a chord diagram by associating each chord with twoExpand
Any nontrivial knot projection with no triple chords has a monogon or a bigon
A generic immersion of a circle into a 2-sphere is often studied as a projection of a knot; it is called a knot projection. A chord diagram is a configuration of paired points on a circle;Expand
Triple chords and strong (1, 2) homotopy
A triple chord is a sub-diagram of a chord diagram that consists of a circle and finitely many chords connecting the preimages for every double point on a spherical curve, and it has exactly threeExpand
Strong and weak (1, 3) homotopies on knot projections
Strong and weak (1, 3) homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two diffe rent types of the third flat Reidemeister moves. InExpand
Strong and weak (1, 2, 3) homotopies on knot projections
A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections ofExpand

References

SHOWING 1-10 OF 20 REFERENCES
On graphs for which every planar immersion lifts to a knotted spatial embedding
We call a graph G intrinsically linkable if there is a way to assign over/under information to any planar immersion of G such that the associated spatial embedding contains a pair of nonsplittablyExpand
Integral geometry of plane curves and knot invariants
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of theExpand
A refinement of the Conway–Gordon theorems
Abstract In 1983, Conway–Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2,Expand
Topological Invariants of Plane Curves and Caustics
Lecture 1: Invariants and discriminants of plane curves Plane curves Legendrian knots Lecture 2: Symplectic and contact topology of caustics and wave fronts, and Sturm theory Singularities ofExpand
Invariants of curves and fronts via Gauss diagrams
Abstract We use a notion of chord diagrams to define their representations in Gauss diagrams of plane curves. This enables us to obtain invariants of generic plane and spherical curves in aExpand
Knots and links in spatial graphs
TLDR
It is shown that any embedding of K7 in three-dimensional euclidean space contains a knotted cycle and that any embedded cycle of K6 contains a pair of disjoint cycles which are homologically linked. Expand
Intrinsic linking and knotting of graphs in arbitrary 3–manifolds
are intrinsically linked, and used these two results to provethat any graph with a minor in the Petersen family (Figure 1) is intrinsically linked.Conversely, Sachs conjectured that any graph whichExpand
Sachs' Linkless Embedding Conjecture
TLDR
It is proved that Sachs′ conjecture that a graph can be embedded in 3-space so that it contains no non-trivial link if and only if it contains as a minor none of the seven graphs obtainable from K 6 by Y − Δ and Δ − Y exchanges. Expand
AN INTRINSIC NON-TRIVIALITY OF GRAPHS
We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsicallyExpand
Intrinsically knotted graphs
In 1983, Conway and Gordon [J Graph Theory 7 (1983), 445–453] showed that every (tame) spatial embedding of K7, the complete graph on 7 vertices, contains a knotted cycle. In this paper, we adapt theExpand
...
1
2
...