[Paper Review] Virtual knot theory on a group

Arnaud Mortier|arXiv (Cornell University)|Mar 13, 2014

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper introduces a unified framework for virtual knot theory using Gauss diagrams decorated by elements of a group equipped with a Z/2-valued homomorphism, generalizing classical and virtual knot invariants. It establishes a 1-1 correspondence between such diagrams and virtual knot diagrams on arbitrary surfaces, generalizes Grishanov-Vassiliev invariants, and defines a Whitney index for non nullhomotopic virtual knots via symmetry-preserving maps and invariance criteria for finite-type invariants.

ABSTRACT

Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all of the earlier attempts to decorate Gauss diagrams, in a way that is made precise via symmetry-preserving maps. These maps become crucial when one makes use of decorated Gauss diagrams to describe finite-type invariants. In particular they allow us to generalize Grishanov-Vassiliev's formulas and to show that they define invariants of virtual knots.

Motivation & Objective

• To develop a general framework for virtual knot theory on arbitrary surfaces using group-valued decorations on Gauss diagrams.
• To unify and generalize previous attempts to decorate Gauss diagrams with topological data, such as homology classes or surface structures.
• To establish a 1-1 correspondence between Gauss diagrams with group decorations and virtual knot diagrams on surfaces, preserving Reidemeister moves.
• To generalize Grishanov-Vassiliev formulas for finite-type invariants in this new framework.
• To define a Whitney index for non nullhomotopic virtual knots using invariants derived from h₁-decorated diagrams.

Proposed method

• Define virtual knot diagrams on an arbitrary surface Σ as tetravalent graphs with real crossings pushed into a real line bundle over Σ.
• Introduce Gauss diagrams decorated by elements of a group π, subject to Reidemeister moves and additional 'conjugacy moves' determined by a Z/2-valued homomorphism w:π→F₂.
• Establish a 1-1 correspondence between such decorated Gauss diagrams and virtual knot diagrams on Σ when π=π₁(Σ) and w=w₁(Σ).
• Introduce abelian Gauss diagrams when π is abelian and w is trivial, showing equivalence to the general framework under these conditions.
• Use symmetry-preserving injections to generalize finite-type invariants, particularly extending Goussarov-Polyak-Viro invariants to this setting.
• Define invariance criteria for w-orbits and apply them to construct new invariants, including the Whitney index and writhe number for non nullhomotopic virtual knots.

Experimental results

Research questions

• RQ1 How can Gauss diagrams be generalized to encode virtual knots on arbitrary surfaces using group-valued decorations and a Z/2-valued homomorphism?
• RQ2 What conditions ensure that the decorated Gauss diagram theory faithfully encodes virtual knot types on a surface Σ?
• RQ3 How can finite-type invariants be generalized in this framework, and what role do symmetry-preserving maps play in this generalization?
• RQ4 Can a Whitney index be defined for non nullhomotopic virtual knots, and how does it relate to classical invariants?
• RQ5 How do the invariants v_l and v_r, derived from h₁-decorated diagrams, behave under Reidemeister moves and what do they represent?

Key findings

• A 1-1 correspondence is established between Gauss diagrams with group decorations (satisfying w:π→F₂) and virtual knot diagrams on a surface Σ when π=π₁(Σ) and w=w₁(Σ).
• The theory generalizes earlier approaches to decorated Gauss diagrams, including those using H₁(Σ)-decorations, via symmetry-preserving maps.
• Grishanov-Vassiliev’s formulas for planar chain invariants are generalized to virtual knots using the proposed framework.
• A Whitney index is defined for non nullhomotopic virtual knots as v_r - v_l, which behaves like the classical Whitney index under Reidemeister moves.
• The sum v_r + v_l is shown to be a virtual knot invariant, analogous to the total writhe number.
• The invariance of v_l and v_r under R₂ and R₃ moves is proven via the w-invariance criterion, and their behavior under R-I moves is fully characterized by a table of changes.