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Genera of two-component alternating links

Noboru Ito, Nodoka Kawajiri

TL;DR

This work extends equality-type results relating the splice-unknotting number to both non-orientable and orientable genera from prime alternating knots to non-splittable two-component alternating links. By developing a splice-based framework with invariants $u^-(L)$ and $u^-_2(L)$ and a unified genus measure $\min\{C(L),\;2g(L)\}$, the authors connect diagrammatic splice sequences to maximal Euler characteristics of spanning surfaces. The main finding is the explicit formula $\min\{C(L),\;2g(L)\}=u^-(L)-[m]=u^-_2(L)-1-[m]$, under the presence of an inter-component $2$-gon, along with a Clark-type inequality extension to links and practical computation via Adams–Kindred state surfaces. The results are supported by a detailed case analysis (prime/non-prime, $u^-$ versus $u^-_2$), a unifying counting framework, and illustrative tables and examples. This advances the understanding of how two-component link genera are governed by splice-unknotting data, with potential implications for algorithmic genus computations in link theory.

Abstract

We extend the equality-type results of Ito--Takimura and Kindred for the non-orientable genera of alternating knots to the setting of two-component alternating links. We show that, for such links, a unified quantity capturing both orientable and non-orientable genera is completely determined by the splice sequence realizing the splice-unknotting number up to an explicit correction term.

Genera of two-component alternating links

TL;DR

This work extends equality-type results relating the splice-unknotting number to both non-orientable and orientable genera from prime alternating knots to non-splittable two-component alternating links. By developing a splice-based framework with invariants and and a unified genus measure , the authors connect diagrammatic splice sequences to maximal Euler characteristics of spanning surfaces. The main finding is the explicit formula , under the presence of an inter-component -gon, along with a Clark-type inequality extension to links and practical computation via Adams–Kindred state surfaces. The results are supported by a detailed case analysis (prime/non-prime, versus ), a unifying counting framework, and illustrative tables and examples. This advances the understanding of how two-component link genera are governed by splice-unknotting data, with potential implications for algorithmic genus computations in link theory.

Abstract

We extend the equality-type results of Ito--Takimura and Kindred for the non-orientable genera of alternating knots to the setting of two-component alternating links. We show that, for such links, a unified quantity capturing both orientable and non-orientable genera is completely determined by the splice sequence realizing the splice-unknotting number up to an explicit correction term.
Paper Structure (13 sections, 9 theorems, 45 equations, 12 figures, 1 table)

This paper contains 13 sections, 9 theorems, 45 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Main result
  4. A proof of Proposition \ref{['Subprop']}
  5. Common framework for proofs
  6. Counting identities.
  7. Clark's inequality
  8. Case $u^-$ (prime)
  9. Case: $u^-$ (non-prime)
  10. Case: $u^-_2$
  11. Proof of Theorem \ref{['Submainthm']}
  12. Table
  13. Examples

Key Result

Theorem 1

Let $L$ be a non-splittable two-component alternating link and let $c$ be a crossing between the two components of an alternating link diagram $D$ of $L$. Let $K$ and $K'$ be knots obtained from splices of two different types on $c$. Suppose that the maximal Euler characteristic of spanning surfaces

Figures (12)

  • Figure 1: Splices
  • Figure 2: (a) type $S^-$ or $S^-_{\textup{join}}$, (b) type $T_{\textup{split}}$, (c) type $T_{\textup{split}}$ corresponding to $\operatorname{RI}^-$, (d) type $\operatorname{RI}^-$
  • Figure 3: $D, S_u, \Sigma_u$
  • Figure 4: Splices used in Adams-Kindred algorithm
  • Figure 5: Deformation $B$
  • ...and 7 more figures

Theorems & Definitions (30)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Theorem 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 20 more