Delta-Unknotting Number for Pretzel Knots

TL;DR

This work determines the Δ-unknotting numbers for important classes of pretzel knots by linking $u^{\Delta}$ to the Conway polynomial coefficient $a_2$. For positive pretzel knots of odd type, and for even type under specific parity conditions, they establish $u^{\Delta}(P)=a_2(P)$ with explicit closed-form expressions, validating these via linking-number calculations and Δ-move techniques. The paper also treats the family $P(-1,p_2,\dots,p_n)$ with odd $n$, deriving a closed form for $u^{\Delta}$ that matches $a_2(P)$, and discusses cases where $u^{\Delta}(P)$ can differ from $|a_2(P)|$, providing counterexamples and a positive odd-type result up to nine crossings. Overall, the results extend the class of knots for which the Δ-unknotting number exactly equals the second Conway coefficient and deepen understanding of Δ-move dynamics on 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. Furthermore, we 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.

Figures (5)

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Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.2
• Lemma 1.2
• Theorem 1.3
• Claim 2.1: MaN
• Claim 2.2: NaNaU
• Claim 2.3
• Proposition 2.4
• Proposition 2.5: Oka1