Delta-Unknotting Number for Pretzel Knots

TL;DR

This paper computes the delta-unknotting number for certain pretzel knots, establishing explicit formulas and extending previous results to new classes of pretzel knots.

Contribution

It provides explicit calculations of the delta-unknotting number for positive pretzel knots and specific non-positive pretzel knots, expanding understanding of their unknotting properties.

Findings

Delta-unknotting number equals the second coefficient of Conway polynomial for positive pretzel knots.

Explicit formulas for delta-unknotting number of pretzel knots of type P(-1, p_2, ..., p_n).

Extended results to pretzel knots with mixed signs and odd parameters.

Abstract

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $\Delta$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $\Delta$-unknotting number for positive pretzel knots. Furthermore, we determine the $\Delta$-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.