This page is unpublished research, intended in the end to become a paper.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.

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.

Terminology

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 $K$.

Definition

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 $w$.

Last revised on August 27, 2018 at 19:47:25.
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