Surgery untying of coloured knots

Abstract

For $p=3$ and for $p=5$ we prove that there are exactly $p$ equivalence classes of $p$–coloured knots modulo $\pm 1$–framed surgeries along unknots in the kernel of a $p$–colouring. These equivalence classes are represented by connect-sums of $n$ left-hand $\left(p,2\right)$–torus knots with a given colouring when $n=1,2,\dots ,p$. This gives a $3$–colour and a $5$–colour analogue of the surgery presentation of a knot.