[Paper Review] Braid presentation of virtual knots and welded knots

Seiichi Kamada|ArXiv.org|Aug 14, 2000

Geometric and Algebraic Topology33 references19 citations

TL;DR

This paper establishes a virtual braid presentation for virtual knots and welded knots, proving that every virtual (or welded) link is the closure of a virtual (or welded) braid, with uniqueness up to specific moves. The key result generalizes the classical Alexander and Markov theorems to virtual and welded settings, introducing virtual exchange moves as essential for equivalence, which do not follow from standard Markov moves in the virtual case.

ABSTRACT

The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.

Motivation & Objective

• To extend the classical braid-theoretic framework of knot theory to virtual and welded knots.
• To resolve a question posed by Kauffman on whether a Markov-type theorem exists for virtual knots.
• To establish a one-to-one correspondence between virtual (or welded) links and virtual (or welded) braids modulo specific moves.
• To clarify the role of exchange moves in virtual braid equivalence, showing they are not derivable from standard Markov moves.

Proposed method

• Introduces virtual braids as an extension of classical braids, defined via the virtual braid group $VB_m$.
• Defines four basic moves: VM0 (braid relations), VM1 (conjugation), VM2 (right stabilization), and VM3 (virtual exchange move).
• Proves that two virtual braids yield equivalent virtual links if and only if they are related by a finite sequence of VM0–VM3 moves.
• Applies the same framework to welded braids, defining WM0–WM2 moves and proving equivalence for welded links.
• Uses Gauss diagrams and isotopy to relate link diagrams to braided diagrams, preserving Gauss data.
• Employs braiding processes to transform link diagrams into braided forms, ensuring consistency under moves.

Experimental results

Research questions

• RQ1 Can every virtual link be represented as the closure of a virtual braid?
• RQ2 What moves are necessary and sufficient to relate virtual braids whose closures yield equivalent virtual links?
• RQ3 Are virtual exchange moves (VM3) redundant in the virtual braid setting, or are they essential?
• RQ4 Can the classical Markov theorem be generalized to welded knots and braids?
• RQ5 How do the moves in the welded braid group compare to those in the virtual braid group?

Key findings

• Every virtual link is the closure of a virtual braid, generalizing the classical Alexander theorem.
• Two virtual braids yield equivalent virtual links if and only if they are related by VM0, VM1, VM2, and VM3 moves.
• The virtual exchange move (VM3) is not derivable from VM0, VM1, and VM2 moves, making it essential in the virtual setting.
• Left stabilizations of positive or negative type in virtual braids cannot be realized using only VM0, VM1, and VM2 moves.
• For welded knots, the closure equivalence is characterized by WM0 (welded braid relations), WM1 (conjugation), and WM2 (right stabilization) moves.
• The same result holds for welded links, with the moves WM0–WM2 providing a complete set of equivalence relations for welded braid closures.