# About trivial torus-knot $T(p,q)$ such that $p$ or $q$ is one

Let $T(p,q)$ be a torus-knot where $p$ and $q$ are coprime. I am asking about the well known fact Which says if $p$ or $q$ is one, then $T(p,q)$ is trivial. If one of $p$ or $q$ is one, then we obtain a knot with crossings each of which can be eliminated by Reidemeister move 1. So at the end the knot is trivial. But I am wondering if we can apply Reidemeister move 1 in a Torus. The torus knot is a circle embedded in a standard torus. In fact $T(1,p)$ is a closed curve with $p$ number of full twists. I am wondering if we can apply Reidemeister move 1 to untwist the knot which is embedded in a torus. If it is embedded in 3-space, I agree. But I can not convinced my self how to deform it to a trivial such that all the deformations are done on a torus.

Any $p,q$ curve on the torus can be thought of as a representative of the homology class $(p,q) \in \mathbb{Z} \oplus \mathbb{Z} \approx H_1(\text{torus})$, so no, it cannot be deformed to a trivial knot within the torus.