[Paper Review] Alexander and Markov theorems for virtual doodles

Neha Nanda, Mahender Singh|arXiv (Cornell University)|Jun 12, 2020

Geometric and Algebraic Topology19 references7 citations

TL;DR

This paper establishes virtual analogues of the Alexander and Markov theorems for doodles on closed oriented surfaces of arbitrary genus. It introduces virtual twin groups as the fundamental algebraic structures generalizing twin groups, proving that every virtual doodle is the closure of a virtual twin and that virtual doodle isotopy classes correspond to virtual twin conjugacy classes under Markov moves, extending classical results to higher genus surfaces.

ABSTRACT

Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to classical knot theory. Alexander and Markov theorems for the genus zero case are known where the role of groups is played by twin groups, a class of right angled Coxeter groups with only far commutativity relations. The purpose of this paper is to prove Alexander and Markov theorems for higher genus case where the role of groups is played by a new class of groups called virtual twin groups which extends twin groups in a natural way.

Motivation & Objective

• Extend classical Alexander and Markov theorems for doodles on the 2-sphere to higher genus surfaces.
• Introduce virtual twin groups as a natural generalization of twin groups to model isotopy classes of virtual doodles.
• Establish a topological realization of virtual twins via braiding processes in the plane.
• Define Gauss data for virtual doodle diagrams and relate them to virtual twin representations.
• Prove that virtual doodle isotopy classes are in bijection with conjugacy classes of virtual twins under Markov moves.

Proposed method

• Define virtual twin groups $VT_n$ as abstract extensions of twin groups $T_n$, incorporating additional generators $ ho_i$ to model virtual crossings.
• Construct a topological model of virtual twins as configurations of $n$ arcs in $\mathbb{R} \times [0,1]$ with virtual crossings allowed.
• Use a braiding process to convert any virtual doodle diagram into a closed virtual twin diagram, preserving Gauss data up to isotopy.
• Apply a sequence of virtual Reidemeister moves ($VR_1, VR_2, VR_3$, $MVR_1$, $MVR_2$) and classical moves ($R_1, R_2$) to relate diagrams.
• Prove that two virtual twin diagrams represent isotopic virtual doodles if and only if their corresponding virtual twins are related by $M0$–$M5$ moves.
• Leverage the surjection from $VT_n$ onto the symmetric group $S_n$ to define the pure virtual twin group as its kernel, analogous to the pure twin group.

Experimental results

Research questions

• RQ1 Can the Alexander theorem for doodles on the 2-sphere be generalized to higher genus surfaces?
• RQ2 What algebraic structure replaces the twin group in the higher genus case for virtual doodles?
• RQ3 How do virtual crossings affect the isotopy classification of doodles on surfaces of genus $g \geq 1$?
• RQ4 Is there a Markov-type theorem for virtual doodles analogous to the classical one for virtual knots?
• RQ5 Can virtual doodle isotopy classes be fully characterized by conjugacy classes of virtual twins?

Key findings

• The virtual twin group $VT_n$ is introduced as a natural extension of the twin group $T_n$, incorporating virtual crossings via generators $\rho_i$.
• Every virtual doodle on a closed oriented surface of genus $g$ is the closure of a virtual twin, generalizing the classical Alexander theorem.
• Two virtual doodles are isotopic if and only if their corresponding virtual twin diagrams are related by $M0$–$M5$ moves, establishing a virtual Markov theorem.
• The Gauss data of a virtual doodle diagram determines the corresponding virtual twin up to $M0$ and $M2$ moves, ensuring algebraic invariance.
• The braiding process converts any virtual doodle diagram into a closed virtual twin diagram while preserving isotopy class, enabling algebraic classification.
• The pure virtual twin group, defined as the kernel of the surjection $VT_n \to S_n$, plays a role analogous to the pure twin group in the classical case.