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

TL;DR

This paper explores the existence of specific plane curves within immersed complete graphs, establishing conditions under which certain chord diagrams appear and analyzing invariants of Hamiltonian plane curves.

Contribution

It proves that large enough complete graphs contain plane curves with prescribed chord diagrams and analyzes invariant sums for immersions of the complete graph on six vertices.

Findings

Existence of plane curves with given chord diagrams in large complete graphs.

Sum of invariants for Hamiltonian plane curves in K6 is congruent to 1/4 mod 1/2.

Any generic immersion of sufficiently large complete graphs contains specific sub-chord diagrams.

Abstract

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.