Virtual Jones Polynomial Overview

• Virtual Jones Polynomial is an extension of the classical Jones polynomial, defined for virtual knots using state-sum and skein-theoretic approaches.
• The invariant employs multiple frameworks such as the Kauffman–Jones formalism, Gauss diagram projections, and surface-state models to unify classical and virtual knot theory.
• Its computational techniques enable detection of non-alternation, orientation, and provide obstructions for local moves, supporting advanced classification in knot theory.

The Virtual Jones Polynomial is an extension of the classical Jones polynomial to virtual link theory, providing a powerful quantum invariant for virtual knots and links. It generalizes the state-sum and skein-theoretic constructions of the Jones polynomial, accommodates Gauss diagram and surface-theoretic formulations, and incorporates invariance under both classical and virtual Reidemeister moves. Multiple formalisms—including the Kauffman–Jones bracket, Gauss diagram projections, surface-state models, and combinatorial expansions—support explicit computation and unify virtual knot invariants with classical and graphical knot theory.

1. State-Sum and Kauffman–Jones Formalism

The most ubiquitous definition for the Virtual Jones Polynomial $f_L(A)$ is through the Kauffman-type state sum, adapted for virtual links. Given an oriented virtual link diagram $L$ with $n$ classical crossings and arbitrary virtual crossings, the state set $S_L$ consists of all $2^n$ choices of resolving each classical crossing as an $A$-splitting or $A^{-1}$-splitting. Virtual crossings are not resolved and remain as 4-valent vertices, imparting no local weight or change to the number of components in the state graph. For each state $s$:

• $a(s)$: number of $A$-resolutions,
• $L$0: number of $L$1-resolutions,
• $L$2: number of loops after all classical crossings are resolved.

The state-sum is then

$L$3

where $L$4 is the writhe (the signed sum over all classical crossings) (Wan, 2017). This formula specializes to the classical Jones polynomial for links without virtual crossings upon setting $L$5.

The state-generating method extends to infinite families of virtual knots, as in the example of the family $L$6, for which closed-form polynomials and linear recurrences can be constructed, allowing concrete computation and detection of non-alternation via “breadth” arguments in the polynomial (Wan, 2017). The method also supports the production of linear recurrences for bracket-state contributions indexed by loop number and combinatorial “sector.”

2. Gauss Diagram and Pseudolink Constructions

In the context of long virtual knots, the virtual Jones polynomial can be defined via explicit projections on Gauss diagrams. Ito's construction employs three maps:

• $L$7: flattening, forgetting over/under data at real crossings,
• $L$8: re-arrowing, turning flat double points into signed undirected chords,
• $L$9: pseudolink projection per Turaev.

The virtual Jones polynomial is then given by

$n$0

where $n$1 is any Gauss diagram of the long virtual knot $n$2 with base point (Ito, 2020). This invariant, strictly stronger than the classical Jones polynomial, distinguishes orientations and mirror images that the classical invariant cannot.

Variants arise from involutions on sign and type of chords, generating a 4-tuple of Jones-type invariants. These are also sensitive to operations such as virtualization, rendering $n$3 robust for virtual knot discrimination.

3. Surface-State and Jones–Krushkal Generalizations

The Jones–Krushkal polynomial $n$4 generalizes the Jones polynomial to diagrams on closed surfaces $n$5, incorporating both combinatorial and homological information:

$n$6

where

• $n$7: states on diagram $n$8 on $n$9
• $S_L$0: numbers of A/B smoothings,
• $S_L$1: homological quantities from inclusion-induced maps on $S_L$2.

The classical and virtual Jones polynomials are recovered as specializations, with $S_L$3 corresponding to the original virtual case. This state-sum satisfies skein and virtualization axioms, and for alternating diagrams on minimum-genus surfaces, reflects important topological minimality and span results (supporting analogues of the Tait conjectures for virtual links) (Boden et al., 2019).

4. Tangle Decompositions and Divisibility Phenomena

Through tangle-theoretic decompositions, the Jones polynomial of oriented virtual links can be expanded in terms of closures of tangles, producing Laurent polynomial coefficients associated with “plug-in” closures. For $S_L$4, the main result is: $S_L$5 where $S_L$6 indexes 2-equal matchings. This decomposition supports general divisibility criteria: the difference $S_L$7 for links differing by a local tangle move is divisible by the greatest common divisor over closure differences. This yields obstructions for local moves (classical crossing change, 4-move, double-4-move), and a necessary condition for S-equivalence of classical knots based on the double-4-move divisor (Drube et al., 2019).

5. Graph-Theoretic (Euler Circuit) Formulation

The Kauffman–Jones polynomial admits a graph-theoretic expansion for checkerboard-colorable 4-valent virtual graphs via the sum over Euler circuits:

• Assign orientation data (source–target structures) and checkerboard coloring.
• For each Euler circuit, construct the chord diagram and evaluate activity words and vertex weights.
• The state-sum $S_L$8 aggregates products of these weights over all Euler circuits.

The normalized Jones–Kauffman polynomial for an oriented link $S_L$9 with shadow graph $2^n$0 is

$2^n$1

This approach affords direct contraction–deletion relations analogous to the skein calculus and admits independence from coloring and labeling, as well as compatibility with classical evaluations and dualities (Abchir et al., 2024). The method parallels classical expansions such as the Thistlethwaite spanning-tree formula but adapts interlacement via chord diagrams for the virtual context.

6. Dominance Properties and Picture Invariants

The label-bracket formalism enhances the expressive power of virtual invariants by encoding resolutions in a module of labeled graphs subjected to relations (instead of mere coefficients). In this language, the Virtual Jones Polynomial emerges as a specialization of the arrow polynomial (via the “normalized arrow polynomial” construction), and the entire label-bracket [D] is strictly stronger than both the classical Jones and arrow polynomials. This formalism ensures functoriality under isotopy and encompasses the Kuperberg $2^n$2 bracket (Akimova et al., 2019).

Evaluations correspond to collapsing thin-edge and vertex data appropriately, and explicit examples recover known values on virtual and classical trefoils. This approach thus unifies and generalizes the quantum invariants of virtual, classical, and higher-graphical knot theory in one combinatorial framework.

7. Applications, Implications, and Extensions

The Virtual Jones Polynomial provides effective tools for distinguishing virtual knots and links, detecting non-alternation and orientation in virtual knots, providing computational obstructions to local moves, and supporting the definition and detection of S-equivalence in classical knot theory through divisibility tests. The formalism extends to:

• Infinite families, allowing explicit closed-form and recurrence computation,
• Surface link theory, capturing both planar and non-planar embeddings,
• Graph-theoretic expansions, relating knot invariants to combinatorics of virtual graphs,
• Dominance frameworks in the module of labeled pictures, supporting extensions to categorifications and to quantum invariants of links in thickened surfaces (Wan, 2017, Ito, 2020, Drube et al., 2019, Boden et al., 2019, Abchir et al., 2024, Akimova et al., 2019).

Extensions remain active in areas such as virtual link homology, categorification, and the study of non-checkerboard-colorable links, as well as the development of further invariants generalizing the Virtual Jones paradigm.