Towards a diagrammatic proof of the Poincaré conjecture for knots (Rev #4)

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\section{Introduction}

This page is unpublished research, intended in the end to become a paper. It aims to give a proof of the restriction of the Poincaré conjecture in its diagrammatic reformulation to the case of a knot diagram.

\section{Realisability of a knot diagram}

The key innovation of our approach is the introduction of a notion of *realisability* of a knot diagram. In this section, we introduce it, after some preliminaries.

\begin{terminology} \label{TerminologyWordInTheArcsOfAKnot} Let $K$ be a knot diagram. Label the arcs of $K$. A *word in the arcs* of $K$ is a monomial $a_{1}^{\pm 1} \cdots a_{n}^{\pm 1}$, where $a_{1}, \ldots, a_{n}$ are labels of arcs of $K$. \end{terminology}

\begin{defn} Let $K$ be a knot diagram, equipped with an orientation (either will do). Let $p$ be a point of $K$. Let $l$ denote the longitude of $K$ with respect to $p$ and our chosen orientation. Let $w$ be a word in the arcs of $K$. A *reduction* of $w$ is a word in the arcs of $K$ obtained by removing a copy of $l$ from $w$. \end{defn}