changing one crossing in (2,n) torus knot

Question

I want to check whether my guess is true or not: changing one crossing in torus knot (2,n) gives a torus knot (2,n-2)? Is that true

By changing crossing I mean exchanging over and under arcs.

Thank you in advance

Answer 1

I'll use braid notation with the usual generating set as it's particularly well suited for this problem.

Theorem. The $(p,q)$-torus knot is equal to the closure of the braid $\beta$ on $p$ strands written as $\beta=(\sigma_1\sigma_2\ldots\sigma_{p-1})^q$.

And so the $(2,n)$-torus knot $K$ is the closure of the braid $\sigma_1^n$.

Switching a crossing in a knot corresponds to replacing the corresponding braid generator in its associated word with its inverse (draw a picture to see this). So if $K'$ is the knot obtained from $K$ by switching a crossing, then $K'$ is the closure of the braid $\beta'=\sigma_1^i\sigma_1^{-1}\sigma_1^j$ for some $i,j\in\mathbb{N}$ such that $i+j+1=n$. It follows that, $$\beta'=\sigma_1^{i-1}\sigma_1^j=\sigma_1^{i+j-1}=\sigma_1^{n-2}$$ which implies that $K'=\overline{\beta'}=(2,n-2)$ as you guessed.

This is a fairly formal proof and has the advantage of using algebra (which is easy) but honestly, the easiest way to see this is to just draw the knot, switch a crossing, and see that this kills off that crossing and one of the adjacent crossings by performing a type II Reidemeister move.