Virtual Knot Theory Overview

Updated 13 December 2025

Virtual knot theory is a generalization of classical knot theory, focusing on knots embedded in thickened surfaces with virtual crossings governed by extended Reidemeister moves.
The theory introduces powerful invariants, including the virtual Jones, Sawollek, and affine index polynomials, to distinguish virtual knots from classical ones.
It bridges diagrammatic, algebraic, and geometric methods, offering new insights into knot concordance, cobordism, and unresolved classification challenges.

Virtual knot theory is a generalization of classical knot theory that extends the study of knots to embeddings in thickened surfaces of arbitrary genus, modulo stabilization. It provides a rich framework unifying diagrammatic, algebraic, geometric, and topological approaches, and has led to new invariants, classification results, and open questions distinct from those in the classical theory.

1. Foundations: Definitions and Diagrammatics

A virtual knot diagram is a 4-valent graph immersed in the plane, with each double point labeled either as a classical crossing (with over/under data) or as a virtual crossing, typically depicted as a circled crossing. Virtual knot diagrams are considered equivalent if they are related by the extended Reidemeister moves, which include:

The three classical Reidemeister moves applied to classical crossings.
The virtual Reidemeister moves, which govern local changes involving only virtual crossings.
The detour move, permitting any segment containing only virtual crossings to be replaced by another such segment, generating only virtual crossings at new intersections.
Forbidden moves are generally not allowed; these include exchanges between classical and virtual crossings in certain configurations (such as the interchange of a classical and a virtual crossing in adjacent positions).

This equivalence is extended to links, knotoids (open-ended diagrams), and other variations such as multi-virtual and flat-virtual knots (Kauffman, 2011, Fenn et al., 2014, Manturov et al., 17 Mar 2024, Kauffman, 10 Sep 2024).

Topologically, Kuperberg's theorem establishes a bijection between virtual knot diagrams modulo virtual Reidemeister moves and knots embedded in thickened surfaces (compact, orientable surfaces without boundary) up to addition and removal of empty handles—a process known as stabilization (Fenn et al., 2014).

2. Algebraic and Polynomial Invariants

The theory has produced a variety of invariants, many extending or generalizing classical knot invariants:

Virtual Jones Polynomial: Derived from a state-sum model, treating classical crossings as usual and virtual crossings as artifacts carried along by the detour move. The polynomial coincides with the classical Jones polynomial for diagrams with no virtual crossings but distinguishes many virtual knots undetectable by classical invariants (Fenn et al., 2014, Kauffman, 2011).
Generalized Alexander Polynomial (Sawollek polynomial): Constructed from the Alexander biquandle, an algebraic structure over a two-variable Laurent polynomial ring ℤ[s^{±1}, t^{±1}]. Its computation proceeds via a module presentation extracted from the diagram, with elementary ideals providing a hierarchy of invariants; the principal generator for knots is the Sawollek polynomial, which vanishes on classical knots but distinguishes virtual knots. An enhancement via Gröbner bases yields finer invariants (Crans et al., 2011).
Affine Index Polynomial and Generalizations: The affine index polynomial and its two-variable generalizations (L^n_K(t, ℓ) and F^n_K(t, ℓ)) detect subtle phenomena associated with virtual crossings and resolve questions on cosmetic crossings and deformations generic to virtual knot diagrams (Kaur et al., 2018).
Bracket and Arrow Polynomials: Extensions of the Kauffman bracket and Jones polynomial using oriented state sums and parity, leading to powerful invariants like the arrow polynomial and its categorifications (Kauffman, 2011). These are sensitive to virtual structures undetected by standard polynomials.

Several of these invariants extend to "flat virtual knots," where all over/under information is suppressed, yielding further generalized Alexander-like and bracket invariants (Manturov et al., 17 Mar 2024).

3. Group-Theoretic and Surface Realizations

Virtual knots admit rich group-valued invariants, many derived from their realization in thickened surfaces:

Virtual Knot Group: Various generalizations of the Wirtinger presentation yield group invariants sensitive to both classical and virtual structure. The virtual knot group, quandle group, and extended Silver–Williams group provide a commutative hierarchy of group-valued invariants, with almost classical knots (those admitting an Alexander numbering) satisfying specific free-product decompositions (Boden et al., 2015).
Lower Central Series: Recent work demonstrates that virtual-knot groups often possess lower central series of greater length than for classical knots, with residual nilpotence and halting stages at transfinite ordinals. These quotients provide powerful invariants: for example, five-step nilpotent quotients can distinguish virtual knots indistinguishable by classical group invariants (Bardakov et al., 2018).
Seifert Surfaces and Alexander Invariants: For almost classical knots (those bounding Seifert surfaces in their minimal genus supporting surface), the Alexander module and polynomial may be constructed analogously to the classical case, but with subtle modifications due to nontrivial surface topology (Boden et al., 2015, Chrisman et al., 2017).

4. Geometric Models and Topos-Theoretic Foundations

Virtual knot theory has been reframed using modern geometric and categorical language:

Grothendieck Topos Model: The space of virtual knots may be described as the category of sheaves on a Grothendieck site built from all surfaces, their knot spaces, and isotopy data. In this model:
- Points correspond to virtual knots (as stabilized embeddings in some thickened surface).
- Paths correspond to virtual isotopies.
- Invariants are morphisms of toposes.
- The inclusion of classical knot theory into virtual knot theory is realized as a specific geometric morphism.
- This formalism rigorously unifies the diagrammatic and topological perspectives and clarifies the embedding of classical knots as a subtheory (Chrisman, 2023).
Virtual Mosaic Theory: Another approach encodes (virtual) knots as combinatorial mosaics on grids, where virtual crossings are represented by special tiles. Mosaic presentations can be mapped onto surfaces of varying genus, with moves corresponding to the Reidemeister and detour moves, as well as surface isotopies and stabilizations. The virtual mosaic number provides a quantitative measure of diagrammatic complexity blending grid size and surface genus (Ganzell et al., 2020).

5. Concordance, Cobordism, and Four-Dimensional Topology

The extension of cobordism theory to the virtual context enriches both knot concordance and surface-based invariants:

Virtual Cobordism: Virtual knots are cobordant if they can be related via a sequence of virtual isotopy moves, births, deaths, and saddle moves. This leads to notions of virtual slice knots (bounding a genus zero surface) and a virtual four-ball genus, which can be calculated via adapted Seifert surface constructions and Turaev’s graded genus invariant (Kauffman, 2014, Boden et al., 2017).
Rasmussen and Lee Invariants: The Lee–homology spectral sequence and extended Rasmussen invariants provide lower bounds on the four-ball genus for positive virtual knots and help detect sliceness obstructions not accessible to classical invariants (Kauffman, 2014, Boden et al., 2017).
Open questions: The classification of the virtual concordance group, the ribbon-slice problem, and the interplay between classical and virtual concordance remain active areas of investigation. Notably, there exist slice virtual knots not pass-equivalent (via band-passing moves) to the unknot, highlighting new phenomena inherent to the virtual category (Kauffman, 2014, Boden et al., 2017).

6. Extensions: Multi-Virtual, Flat, and Free Knots

Recent developments have produced further generalizations:

Multi-Virtual Knot Theory: This extension considers diagrams with multiple types of virtual crossings, each with its own detour move and associated combinatorial axioms. The introduction of multiple virtual types corresponds to generalizations of state-sum invariants, including Penrose-type bracket evaluations on generalized graphs. These frameworks admit new polynomial invariants, quandles, and extensions of classical skein-theoretic and quantum techniques (Kauffman, 10 Sep 2024).
Flat and Free Knots: Flat virtual knots are diagrams where classical crossings lose all over/under data. Their invariants arise from Alexander-like polynomials and picture-valued brackets tailored to the flat category (Manturov et al., 17 Mar 2024). Free knots, which are equivalence classes of flat virtual diagrams under parity moves, form a separate area with open classification and sliceness problems (Fenn et al., 2014).

7. Classification Results, Applications, and Open Problems

Virtual knot theory yields classification theorems, algorithmic tools, and a wide range of open questions:

Minimal-genus Realizations: Every virtual knot admits a unique minimal-genus supporting surface (up to homeomorphism and stabilization). This facilitates sharp obstructions to classicality and provides a geometric measure (the minimal genus) for both virtual knots and knotoids (Fenn et al., 2014, Gügümcü et al., 25 Feb 2025).
Slice Genus Computations: Sliceness and four-ball genus can be algorithmically computed for low-crossing knots using parity techniques, Turaev's graded genus, and combinatorial reductions (Boden et al., 2017).
Forbidden Detour Number: The forbidden detour number quantifies the minimal number of forbidden detour moves required to unknot a virtual knot, providing a complexity invariant distinct from both crossing number and polynomial data (Yoshiike et al., 2019).
Unsolved Problems: Foundational challenges include the extension of quantum and homological invariants to large classes of virtual knots, classification of flat and free knots, understanding virtual 3-manifolds, the structure of the virtual braid group, and the search for universal finite-type invariants in the virtual setting (Fenn et al., 2014). The Kishino knot and related phenomena exemplify the subtlety of these issues, being undetectable by many quantum invariants yet distinguished by parity and surface invariants.

Virtual knot theory thus encompasses a comprehensive and rapidly developing area at the intersection of low-dimensional topology, combinatorics, and algebra. It exhibits new invariants and phenomena without classical analogues, and continues to bridge concepts from classical knot theory, 3-manifold topology, quantum algebra, and categorical geometry (Fenn et al., 2014, Chrisman, 2023, Kauffman, 10 Sep 2024).