On compressing discs of torus knots (Q2716945)

A compressing disc for a knot \(K\) is a map \(f:D^2\to S^3\) realizing a null homotopy of \(K=f|_{\partial D^2}\) and such that \(f\) is transverse to \(K\) on the interior of the disc. The knottedness of \(K\) (in the sense of Pannwitz) is the minimum of the numbers of intersections of \(K\) with such compressing discs. If \(S^3\) is oriented we may distinguish between positive and negative intersections. This paper gives a simple direct proof that the knottedness of the \((2n+1,2)\)-torus knot is \(2n\), and that moreover any compressing disc has at least \(2n\) negative intersections with \(K\). The key idea is a diagrammatic representation of null homotopies of a longitude in \(S^3\). It is suggested that the method may apply to certain other 2-bridge knots.