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Unknotting number and connected sums: The knots $4_1$ and $5_1$

Mark Brittenham, Susan Hermiller

TL;DR

The paper addresses the additivity of the unknotting number under connected sums by introducing Gordian adjacency and the concept of symbiont knots, and then constructs explicit counterexamples for $K\in\{4_1,5_1\}$ via $K_\text{sum}=K\#K'$ with $u(K\#K')<u(K)+u(K')$. It also explores a potential symbiont for $3_1$ through $10_6$ and analyzes the open status of $u(10_6)$ to delineate possible outcomes. By combining explicit 15-crossing diagrams, DT-codes, SnapPy verifications, and Bernhard-Jablan unknotting-number data, the work demonstrates that unknotting-number additivity can fail and discusses a broader landscape of symbionts and Gordian adjacency across knots up to 10 crossings. The results motivate a closer look at the mechanisms behind symbiosis in unknotting and outline computational and theoretical directions for identifying further counterexamples and structural patterns.

Abstract

We show that the knots $K\in\{4_1,5_1\}$ can be paired with a corresponding knot $K^\prime$ such that $u(K\#K^\prime)<u(K)+u(K^\prime)$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot $K^\prime$ for the knot $3_1$.

Unknotting number and connected sums: The knots $4_1$ and $5_1$

TL;DR

The paper addresses the additivity of the unknotting number under connected sums by introducing Gordian adjacency and the concept of symbiont knots, and then constructs explicit counterexamples for via with . It also explores a potential symbiont for through and analyzes the open status of to delineate possible outcomes. By combining explicit 15-crossing diagrams, DT-codes, SnapPy verifications, and Bernhard-Jablan unknotting-number data, the work demonstrates that unknotting-number additivity can fail and discusses a broader landscape of symbionts and Gordian adjacency across knots up to 10 crossings. The results motivate a closer look at the mechanisms behind symbiosis in unknotting and outline computational and theoretical directions for identifying further counterexamples and structural patterns.

Abstract

We show that the knots can be paired with a corresponding knot such that . As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot for the knot .
Paper Structure (7 sections, 5 theorems, 5 figures)

This paper contains 7 sections, 5 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Proofs of the Theorems
  3. The unknotting number of $10_6$ and its consequences
  4. Further examples, via Gordian Adjacency
  5. The road forward
  6. Verification code
  7. Acknowledgements

Key Result

Theorem 1.1

The connected sum $K_4=4_1\#9_{10}$, of the knots $4_1$ and $9_{10}$, satisfies $u(4_1)=1$, $u(9_{10})=3$ and $u(4_1\#9_{10})\leq 3$.

Figures (5)

  • Figure 1: Initial diagram for $4_1\#9_{10}$
  • Figure 2: Crossing changes to obtain the knot $8_{14}$
  • Figure 3: From $4_1\#9_{10}$ to the unknot
  • Figure 4: Initial diagram for $5_1\#8_2$
  • Figure 5: Initial diagram for $3_1\#10_6$

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm:four-one']}
  • proof : Proof of Theorem \ref{['thm:five-one']}
  • proof : Proof of Theorem \ref{['thm:three-one']}
  • Lemma 4.1
  • Corollary 4.2
  • Conjecture 5.1