Richard Williamson
Towards a diagrammatic proof of the Poincaré conjecture for knots



This page is unpublished research, intended in the end to become a paper.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 KK be a knot diagram. Label the arcs of KK. A word in the arcs of KK is a monomial a 1 ±1a n ±1a_{1}^{\pm 1} \cdots a_{n}^{\pm 1}, where a 1,,a na_{1}, \ldots, a_{n} are labels of arcs of KK. \end{terminology}

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