Delta-Unknotting Number for Two-Bridge Knots

TL;DR

This paper calculates the minimum number of delta-moves needed to unknot certain classes of two-bridge knots, providing explicit values and discussing cases with a single move.

Contribution

It determines the delta-unknotting numbers for specific classes of two-bridge knots, a novel contribution to knot theory.

Findings

Explicit delta-unknotting numbers for two classes of two-bridge knots

Identification of two-bridge knots with delta-unknotting number one

Enhanced understanding of unknotting operations in knot theory

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. We determine the $\Delta$-unknotting numbers for two-bridge knots of type $C(2\beta_1, 2\beta_2, ... , 2\beta_n)$ and type $C(2\beta_1, 2\beta_2, ... , 2\beta_{n-1}, 2\beta_n-1)$, where $\beta_i$ is a positive integer for $1 \leq i \leq n$. We also discuss two-bridge knots whose $\Delta$-unknotting number is equal to one.