[Paper Review] Virtual strings and their cobordisms

Vladimir Turaev|arXiv (Cornell University)|Nov 12, 2003

Geometric and Algebraic Topology9 references17 citations

TL;DR

This paper introduces virtual strings as combinatorial models of self-intersections of curves on surfaces, developing algebraic invariants—particularly a polynomial $ u $ and based matrices—to study their homotopy and cobordism classes. It establishes that the group of homotopy invariants forms an infinite-dimensional Lie group, and constructs a Lie cobracket on the free abelian group of homotopy classes, linking virtual strings to virtual knots via skein algebras and Hopf algebra structures.

ABSTRACT

A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connections between virtual strings and virtual knots.

Motivation & Objective

• To formalize virtual strings as combinatorial objects representing self-intersections of curves on surfaces.
• To define and study homotopy and cobordism equivalence relations on virtual strings using algebraic invariants.
• To establish a Lie algebra and Lie group structure on the group of homotopy invariants of virtual strings.
• To relate virtual strings to virtual knots via polynomial invariants and skein algebra isomorphisms.
• To develop obstructions to sliceness using the polynomial $ u $ and based matrices, and to explore connections with 3-manifold topology.

Proposed method

• Define a virtual string of rank $ m $ as an oriented circle with $ 2m $ distinct points partitioned into $ m $ ordered pairs (arrows), representing self-intersections.
• Introduce the polynomial invariant $ u $, derived from the based matrix of a string, to detect homotopy and cobordism obstructions.
• Construct a Lie cobracket on the free abelian group generated by homotopy classes of strings, dualizing the Lie algebra structure on homotopy invariants.
• Define cobordism of strings via 3-manifolds with boundary, and introduce the notion of a slice string as one bounding a singular disk in a contractible 3-manifold.
• Use the based matrix $ T(eta) = (G, s, b) $ to define genus and sliceness obstructions, with $ |b(e,f)| \leq \#(G) - 2 $ as a constraint.
• Establish a comodule structure on open strings over the Lie coalgebra of closed strings, enabling group actions via $ \operatorname{Exp}\mathcal{A}^* $ when $ R \supset \mathbb{Q} $.

Experimental results

Research questions

• RQ1 Which primitive based matrices can be realized as $ T_{\bullet}(\alpha) $ for some virtual string $ \alpha $, and what constraints exist?
• RQ2 Can secondary obstructions detect non-slice strings with hyperbolic based matrices?
• RQ3 Are all slice strings stably ribbon, i.e., is the product of a slice string with a ribbon string homotopic to a ribbon string?
• RQ4 Is every virtual string homotopic to a string of type $ \alpha_\sigma $ for some permutation $ \sigma $, or up to cobordism?
• RQ5 Is multiplication of open strings commutative up to homotopy or up to cobordism?

Key findings

• The group of $ \mathbb{Z} $-valued homotopy invariants of virtual strings forms a Lie algebra, which integrates into an infinite-dimensional Lie group.
• The free abelian group generated by homotopy classes of virtual strings carries a natural Lie cobracket, making it a Lie coalgebra.
• The skein algebra of virtual knots is isomorphic to the polynomial algebra generated by homotopy classes of virtual strings, via a map to $ \mathbb{Q}[z] $-coefficients.
• A string is slice if it bounds a singular disk in a 3-manifold that is contractible; obstructions to sliceness are given by the polynomial $ u $ and based matrix invariants.
• The module of open virtual strings becomes a comodule over the Lie coalgebra of closed strings, and when $ R \supset \mathbb{Q} $, the group $ \operatorname{Exp}\mathcal{A}^* $ acts on it by algebra automorphisms.
• The construction yields a homeomorphism invariant of knots in cylinders over surfaces with values in $ \mathbb{Q}[z,t] $, generalizing classical invariants.