Characterization of signed Gauss paragraphs

TL;DR

This paper explores the minimal surface realizations of signed Gauss paragraphs using embedded graph theory, providing new insights into their genus and solutions to the signed Gauss word problem.

Contribution

It introduces a method to construct minimal realizations of signed Gauss paragraphs and relates their genus to Carter's circles, also solving the signed Gauss word problem.

Findings

Minimal realizations of signed Gauss paragraphs are characterized by surface genus.

The genus can be expressed as a function of Carter's circles.

A short solution to the signed Gauss word problem is provided.

Abstract

In this paper we use theory of embedded graphs on oriented and compact $PL$-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter's circles. For the case of signed Gauss words, we use a generating set of $H_1(S_w,\mathbb{Z})$, given in \cite{CaEl}, and the intersection pairing of immersed $PL$-normal curves to present a short solution of the signed Gauss word problem. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus $PL$-surface.