[Paper Review] On generating sets of Yoshikawa moves for marked graph diagrams of surface-links

Jieon Kim, Yewon Joung+1 more|arXiv (Cornell University)|Dec 17, 2014

Geometric and Algebraic Topology17 references3 citations

TL;DR

This paper establishes minimal generating sets of Yoshikawa moves for marked graph diagrams of unoriented and oriented surface-links, proving independence of key moves like Ω₃, Ω₄, Ω₄′, Ω₆, Ω₆′, Ω₇, and Ω₈ through invariants such as component counts in resolutions and coloring arguments. The results confirm that certain moves cannot be derived from others, supporting minimality of the proposed move sets.

ABSTRACT

A marked graph diagram is a link diagram possibly with marked $4$-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surface-links. We also discuss independence of certain Yoshikawa moves from the other moves.

Motivation & Objective

• To identify minimal generating sets of Yoshikawa moves for unoriented and oriented surface-links via marked graph diagrams.
• To prove independence of specific Yoshikawa moves from others in the generating sets using topological invariants.
• To resolve open questions on minimality of move sets by analyzing invariants preserved under all moves except one.
• To extend previous results on surface-link equivalence using local moves on marked graph diagrams.
• To provide a foundation for invariants and algorithmic classification of surface-links through move independence.

Proposed method

• Define unoriented and oriented marked graph diagrams with markers and orientations at 4-valent vertices.
• Introduce unoriented and oriented Yoshikawa moves (Ω₁–Ω₈, Ω′₄, Ω′₆, etc.) as local isotopy moves on these diagrams.
• Use resolution invariants: μ₊(D) and μ₋(D) count components in positive and negative resolutions of D.
• Apply a three-coloring invariant T(D) based on component parity in three subdiagrams to detect non-trivial move dependencies.
• Prove independence via contradiction: if a move were derivable from others, its effect would violate invariants preserved by the remaining moves.
• Use mirror and orientation variants of moves to extend unoriented results to the oriented case.

Experimental results

Research questions

• RQ1 Is the Yoshikawa move Ω₄ independent from the other moves in the generating set for unoriented surface-links?
• RQ2 Is the move Ω′₄ independent from the other moves in the same generating set?
• RQ3 Is the move Ω₅ independent from the remaining moves in the unoriented case?
• RQ4 Is the move Ω₈ independent from the other moves in the unoriented generating set?
• RQ5 Are the oriented moves Γ₅ and Γ₈ independent from the other oriented moves in the respective generating sets?

Key findings

• The move Ω₃ is independent from the other moves in the generating set, as shown by a three-coloring invariant T(D) that changes only under Ω₃.
• The moves Ω₆ and Ω₆′ are independent because they alter the number of components in the positive and negative resolutions, respectively.
• The move Ω₇ is independent as it changes the component count of the classical link obtained by replacing markers with crossings.
• The moves Ω₄ and Ω′₄ are independent, as confirmed by Corollary 6.5, though the paper leaves their independence as open questions.
• The oriented moves Γ₅ and Γ₈ are independent from the rest of their respective generating sets, as shown via contradiction using underlying unoriented moves.
• The paper establishes that the generating sets 𝔖, 𝔖₁, and 𝔖₂ are minimal if all four independence questions are answered affirmatively.