Permutation Jones Polynomials

Abstract: We introduce a generalization of the Jones polynomial for classical and virtual knots and links using colorings by a permutation $σ:X\to X$ of a finite set $X$. For $X={1}$ and for classical knots, the invariant is equivalent to the usual Jones polynomial; for $X$ with cardinality greater than 1 the invariant expresses distinct information from the Jones polynomial or virtual knots and for classical and virtual links. We establish some properties of the new invariants and compute the polynomials for classical and virtual knots and links of small crossing number for a few small permutations.

• The paper introduces permutation Jones polynomials as an extension of the classical Jones polynomial using σ-colorings of knot and link diagrams.
• The paper develops a three-variable skein relation that differentiates polychromatic from monochromatic crossings, enhancing the invariant's power in virtual settings.
• The paper demonstrates through computations that these invariants detect link splitting and virtuality, offering new insights in quantum topology.

Permutation Jones Polynomials: A Generalization of the Jones Polynomial via Permutation Colorings

Introduction

This work introduces a new family of link invariants, termed permutation Jones polynomials, which extend the classical Jones polynomial via colorings derived from a permutation $\sigma: X \to X$ on a finite set. The construction covers both classical and virtual knots and links and, for $|X|=1$, recovers the standard Jones polynomial. However, for $|X|>1$, these invariants encode genuinely distinct information, especially in the context of virtual knots and linked components in classical links. This approach is motivated by connections to biquandle virtual brackets but achieves a particularly transparent combinatorial definition in the constant action case.

Construction of the Invariant

The invariant is built by considering $\sigma$-colorings of knot and link diagrams, where each semiarc in the diagram is labeled by an element of $X$ so that certain consistency relations, determined by the permutation $\sigma$, are satisfied at each crossing. The set of all $\sigma$-colorings is preserved under Reidemeister moves, yielding a well-defined coloring set $\mathcal{C}_\sigma(L)$ for any oriented knot or link $L$. Notably, for classical knots and $|X| > 1$, all crossings are forced to be monochromatic, yielding no new information beyond the classical Jones polynomial; in contrast, for links (and virtual knots/links), both monochromatic and polychromatic crossings can occur, leading to strictly stronger invariants.

A permutation Jones polynomial $|X|=1$0 is then defined as a weighted, normalized state sum over all $|X|=1$1-colorings and all Kauffman bracket states within each coloring, using a three-variable skein relation. At monochromatic crossings, Kauffman bracket-style relations are applied; at polychromatic crossings, a virtualizing operator together with an explicit $|X|=1$2-variable factor is invoked. The total degree of each monomial in $|X|=1$3 is zero due to the specific normalization, and the resulting invariant is strictly more general than the original Jones polynomial.

Fundamental Properties

The invariant satisfies several critical structural properties:

• For classical knots, $|X|=1$4 factors as $|X|=1$5, so no new information is gained aside from multiplicity.
• For classical links, the invariant often distinguishes links that the Jones polynomial cannot, as the presence of polychromatic crossings can introduce $|X|=1$6-terms (which, for instance, are obstructed in split links).
• For virtual knots and links, the behavior is even richer; the $|X|=1$7-variable provides a clear obstruction to planarity, and the invariant can distinguish virtual knots/links with the same standard Jones polynomial.

Permutation Jones polynomials are invariant under mutation and mirror image as expected, but, for nontrivial $|X|=1$8, are not invariant under Kauffman virtualization—a strict contrast with the classical Jones polynomial.

The invariant is also shown to be neither determined by nor determines the standard Jones polynomial for $|X|=1$9, with explicit examples where different virtual knots share the same Jones polynomial but exhibit distinct $|X|>1$0 polynomials, and vice versa.

Computations and Empirical Results

The paper provides numerous explicit computed examples for both classical and virtual knots and links with small crossing numbers and for several nontrivial permutations (e.g., $|X|>1$1, $|X|>1$2).

Key computational observations include:

• For some classical links of small crossing number, $|X|>1$3 can detect nonsplitness by the presence of nonzero $|X|>1$4-terms, even when the Jones polynomial fails to do so.
• For many virtual knots, the structure of $|X|>1$5 records more data about virtuality and coloring than the Jones polynomial, and essential features (for example, which knots admit nontrivial colorings for a given $|X|>1$6) are permutation-dependent.
• The computational complexity for calculating $|X|>1$7 increases with $|X|>1$8 but can, in favorable cases, be mitigated by symmetry in the coloring states.

Theoretical and Practical Implications

The permutation Jones polynomial establishes a natural class of invariants in the context of skein-theoretic and quandle-theoretic approaches to knot theory. The structure suggests several lines of inquiry:

• The invariant, being a particular family of biquandle virtual brackets, positions itself as a bridge between biquandle theory and skein-theoretic invariants. Understanding which biquandle structures admit similar constructions is relevant for the broader program of quantum and algebraic enhancements of knot invariants.
• The capacity to detect link splits (via $|X|>1$9-terms) suggests applications in link classification not addressed by the Jones polynomial or its immediate relatives.
• The dependency of $\sigma$0 on the cycle structure or properties of $\sigma$1 highlights a new avenue for investigating the algebraic underpinnings of knot invariants, potentially analogous to the study of coloring invariants in quandle and biquandle theory.

On the computational side, the presence of multiple colorings for $\sigma$2 increases the number of states but can sometimes allow for aggregation of identical contributions, depending on the permutation’s symmetry. Efficient computation strategies exploiting this could be developed.

Open Questions and Future Directions

The paper concludes with several open questions:

• Can the class of all biquandle virtual brackets with permutation-like skein coefficients be classified or characterized? What additional invariants are derivable in this direction?
• Is the multiset of state-sum contributions (as opposed to their sum) a strictly stronger invariant? Are there explicit links or virtual knots distinguished at the multiset level but not by the summed polynomial?
• Are there Jones-equivalent classical links (as opposed to only virtual knots) that are distinguished for nontrivial $\sigma$3?
• What properties of two permutations $\sigma$4 guarantee $\sigma$5 for certain families of links or knots?
• Can these invariants be further categorified, in analogy to Khovanov homology?

These questions highlight the mathematical and computational fertility of the approach.

Conclusion

Permutation Jones polynomials substantially generalize the Jones polynomial by leveraging permutation-induced colorings and associated skein-theoretic modifications. Their sensitivity to both classical and virtual crossing data, their permutation-dependent structure, and their deep connections to biquandle bracket theory position them as important objects for further study in quantum topology. The results elucidate new avenues for the detection and classification of links and knots, particularly in virtual and multi-component settings, and invite further development in invariant theory and knot categorification.