Delta-Unknotting Number for Pretzel Knots

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. It is known that, for positive pretzel knots, the $\mathrm{\Delta }$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $\mathrm{\Delta }$-unknotting number for positive pretzel knots. As a consequence of the above result, among positive pretzel knots of odd type with a fixed crossing number $n$, where $n$ is odd, the $\mathrm{\Delta }$-unknotting number is maximized by $P(1,1,\mathrm{.}\mathrm{.}\mathrm{.},1)(\cong T(2,n))$, and the maximum value is $\frac{1}{8}(n^2-1)$. We also obtain a similar result for torus knots. We further determine the $\mathrm{\Delta }$-unknotting number for pretzel knots of type $P(-1,p_2,\mathrm{.}\mathrm{.}\mathrm{.},p_n)$, where $p_i$ is a positive odd integer for $2\le i\le n$ and $n$ is odd.