Delta-Unknotting Number for Pretzel Knots

Abstract

The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $Δ$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $Δ$-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 $Δ$-unknotting number is maximized by $P(1, 1, ... , 1) \quad \big( \cong T(2,n) \big)$, and the maximum value is $\frac{1}{8}(n^2 - 1)$. We also obtain a similar result for torus knots. We further determine the $Δ$-unknotting number for pretzel knots of type $P(-1, p_2, ..., p_n)$, where $p_i$ is a positive odd integer for $2 \leq i \leq n$ and $n$ is odd.