[Paper Review] Quotient Quandles and the Fundamental Latin Alexander Quandle

Sam Nelson, Sherilyn Tamagawa|arXiv (Cornell University)|Apr 24, 2014

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper introduces the Fundamental Latin Alexander Quandle (FLAQ) as a new quandle invariant for classical and virtual knots, generalizing the Alexander quandle by extending coefficients to ensure the quandle is Latin. Using Gröbner basis techniques on the resulting quotient, the authors define FLAG invariants that generalize the Alexander polynomial. The key contribution is demonstrating via example that the FLAG invariant for virtual knots is not determined by the generalized Alexander polynomial, showing its potential for distinguishing virtual knots beyond classical invariants.

ABSTRACT

Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, which generalize the Alexander polynomial. We show via example that the invariant is not determined by the generalized Alexander polynomial for virtual knots.

Motivation & Objective

• To develop new quandle-based invariants for classical and virtual knots using quotient structures of the fundamental quandle.
• To generalize the Alexander polynomial via Gröbner basis-valued invariants derived from a new quandle construction.
• To investigate whether these new invariants can detect differences in virtual knots that are indistinguishable by the generalized Alexander polynomial.
• To provide an algorithmic framework for computing these invariants using module-theoretic and computational algebraic techniques.

Proposed method

• Define the Fundamental Latin Alexander Quandle by extending the coefficient ring of the Alexander quandle to ensure the quandle operation satisfies the Latin property (i.e., right multiplication is bijective).
• Construct the FLAG invariant by computing a Gröbner basis of the ideal generated by the quandle relations in the Laurent polynomial ring Z[s±1, t±1].
• Use monomial orderings to standardize the Gröbner basis representation, enabling comparison across knots.
• Apply the FLAG invariant to all classical knots with up to eight crossings, computing |FLAG₁(K)| and the full Gröbner basis for each.
• Extend the FLAG construction to virtual knots by ignoring virtual crossings in the quandle presentation, preserving the algebraic structure.
• Use computational algebra systems to verify invariants and compare results, particularly contrasting FLAG with the generalized Alexander polynomial.

Experimental results

Research questions

• RQ1 Can the FLAG invariant distinguish virtual knots that have the same generalized Alexander polynomial?
• RQ2 What is the relationship between the cardinality of the FLAG₁ invariant and topological or algebraic knot invariants?
• RQ3 Are there other quotient quandles of the fundamental quandle that yield finite, computable, and informative invariants for knots?
• RQ4 How do FLAG invariants relate to existing quandle counting invariants and their enhancements?
• RQ5 What structural properties of the coefficient ring extension ensure the resulting quandle is Latin and yields a non-trivial invariant?

Key findings

• The FLAG₁ invariant for the virtual knot 4.99 is {s⁻¹ - 2, t⁻¹ - 2, 2s - 1, 2t - 1}, which differs from that of the trefoil knot despite both having the same virtual Alexander polynomial.
• The FLAG₁ invariant is not determined by the generalized Alexander polynomial, as demonstrated by the 4.99 virtual knot and the trefoil, which are distinguished by FLAG₁ but not by the polynomial.
• For all classical knots with up to eight crossings, the FLAG₁ invariant was computed and found to have cardinality 7 in all cases, indicating a consistent structure across this class.
• The FLAG invariants are defined for virtual knots by treating virtual crossings as non-interfering, allowing the same algebraic framework to be applied.
• The Fundamental Latin Alexander Quandle is constructed by extending the coefficient ring of the Alexander quandle to ensure the quandle is Latin, enabling stronger algebraic control.
• The Gröbner basis-based FLAG invariants generalize the Alexander polynomial and provide a richer algebraic invariant for virtual knots.