Montesinos Knots with Delta-Unknotting Number One

The $\mathrm{\Delta }$-unknotting number for a knot is defined as the minimum number of $\mathrm{\Delta }$-moves needed to deform the knot into the trivial knot. In this paper, we discuss Montesinos knots whose $\mathrm{\Delta }$-unknotting number is equal to one. We propose a conjectural characterization of Montesinos knots with $\mathrm{\Delta }$-unknotting number one and provide examples admitting distinct $\mathrm{\Delta }$-moves, each of which deforms the knot into the trivial knot.