Biquandle Fares and Link Invariants

We introduce a new family of invariants of oriented classical and virtual knots and links using fares, maps from paths in biquandle-colored diagrams to an abelian coefficient group. We consider the cases of 1-fares and 2-fares, provide examples to show that the enhancements are proper and end with some open questions about the cases of n-fares for n > 2.

💡 Research Summary

The paper introduces a novel family of enhancements for biquandle‑based knot and link invariants, called “biquandle fares.” A fare is a linear map ϕ from the free abelian group generated by n‑tuples of biquandle elements (i.e., paths of length n in the quiver obtained from a diagram) to an abelian coefficient group A. For each colored route one records (−1)^k ϕ(x₁,…,x_n), where k is the number of negative crossings encountered along the route, and sums these contributions over an appropriate collection of routes. The authors derive a set of algebraic conditions—“fare axioms”—that guarantee invariance of the total sum under X‑colored Reidemeister moves.

The simplest case, n = 1, yields 1‑fares: functions ϕ : X → A satisfying three equations (i)–(iii) corresponding respectively to Reidemeister I, II, and III moves. The fare value of a diagram is obtained by replacing each semiarc color x with ϕ(x) and adding over all semiarcs. Collecting these values over the entire biquandle homset Hom(B(L), X) produces a multiset Φ_{ϕ,M}^X(L). The authors encode the multiset as an “additive polynomial” Φ⁺ (where the coefficient of x^k counts colorings with fare value k) and a “multiplicative polynomial” Φ× (whose roots are the fare values). Evaluating Φ⁺ at x = 1 recovers the ordinary biquandle counting invariant, showing that 1‑fares are genuine enhancements.

For n = 2 the paper distinguishes three types of fares:

• Complete 2‑fares consider every possible ordered pair (x, y) that can appear on a length‑2 route.
• Through 2‑fares restrict to routes that follow the over‑ or under‑strand continuously at each crossing.
• Crooked 2‑fares restrict to routes that always turn at a crossing, never proceeding straight.

Each type satisfies a more intricate set of axioms (again labeled (i)–(iii)) derived from checking invariance under the three Reidemeister moves. A complete 2‑fare can be expressed as the sum of a through and a crooked fare, but the converse need not hold; the authors therefore define “decomposable” (sum of through + crooked) and “indecomposable” 2‑fares.

The paper provides extensive computational examples. Using small finite biquandles (e.g., 2‑element, 3‑element, and 4‑element tables) and coefficient groups such as ℤ₅, ℤ₂⊕ℤ₂, and ℤ₆, the authors generate all possible 1‑ and 2‑fares via Python scripts. For each fare they compute the multiset and polynomial invariants for a variety of classical and virtual knots/links (trefoil 3₁, figure‑eight 4₁, Hopf link, virtual Hopf link, and many links up to seven crossings). The results demonstrate that distinct knots with identical biquandle counting values can be distinguished by their fare polynomials. For instance, the trefoil and the unknot both have 16 X‑colorings for a certain 4‑element biquandle, yet the trefoil’s 2‑fare polynomial is x¹⁶ (all fares zero) while the virtual Hopf link yields a non‑trivial multiset {2 × (1,1), 1 × (0,0)}. Similar discriminating power is shown for higher‑crossing links, confirming that fares are proper enhancements.

The authors also discuss the algebraic structure of the fare space: complete 2‑fares form an abelian group under pointwise addition, and the subgroups of through and crooked fares intersect trivially. They note that many complete fares are indecomposable, indicating richer structure beyond simple linear combinations.

In the concluding section, the paper acknowledges that the theory for n > 2 remains undeveloped. Open questions include: (1) how to formulate coherent axioms for higher‑order fares; (2) whether higher‑order fares can detect phenomena invisible to 1‑ and 2‑fares (e.g., subtle virtual knot invariants); (3) what computational complexity arises when enumerating routes of length n; and (4) how fare invariants interact with other enhancements such as quandle cohomology or biquandle brackets. The authors suggest that exploring these directions could further illuminate the relationship between biquandle colorings, path algebras, and knot theory.

Overall, the paper contributes a fresh algebraic framework—biquandle fares—that systematically augments the biquandle counting invariant, provides concrete computational tools, and opens a promising avenue for future research in knot invariants.