Knots in $S_{g} \times S^{1}$, their essential diagrams and virtual knots

TL;DR

This paper explores the minimal diagrams of knots in surface cross circle spaces, demonstrating the embedding of virtual knot theory into this context and analyzing classical links derived from these knots.

Contribution

It introduces the concept of essential diagrams with minimal double lines and shows how virtual knot theory is embedded in knots in $S_{g} imes S^{1}$, also studying classical links from these knots.

Findings

Virtual knot theory embeds into knots in $S_{g} imes S^{1}$.

Minimal diagrams with fewest double lines are characterized.

Classical links from knots in $S^{2} imes S^{1}$ are analyzed for minimality and separability.

Abstract

In \cite{Kim} it is shown that knots in $S_{g} \times S^{1}$ can be presented by virtual diagrams with a decoration, so called, {\em double lines}. In this paper, we study the essential diagram for each knot in $S_{g} \times S^{1}$, which has the minimal number of double lines. We prove that virtual knot theory is embedded in the theory of knots in $S_{g}\times S^{1}$. In the same time, one can obtain knots in $S^{2}\times S^{1}$ from 2-component links $L = K\sqcup T$ where $T$ is a trivial knot. By using knots in $S^{2} \times S^{1}$, we study the minimality and separability of such classical links.