Quadratic quandles and their link invariants

TL;DR

This paper investigates link invariants derived from quadratic Alexander quandles, revealing a specific algebraic form for these invariants and proposing a conjecture linking them to the Alexander module, supported by computational evidence.

Contribution

It introduces a new class of link invariants from quadratic Alexander quandles and conjectures their determination by the Alexander module, supported by computational verification.

Findings

Invariant values have the form Gamma_p^r p^{2s} for fixed Gamma_p and integers r,s.

The invariant is determined by the Alexander module for all torus and two-bridge knots.

Computational evidence supports the conjecture relating the invariant to the Alexander module.

Abstract

Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreducible modulo p. For each such quandle, there is an invariant with values in the group ring Z[C_p] of a cyclic group of order p. We shall show that the values of this invariant all have the form Gamma_p^r p^{2s} for a fixed element Gamma_p of Z[C_p] and integers r >= 0 and s > 0. We also describe some machine computations, which lead us to conjecture that the invariant is determined by the Alexander module of the link. This conjecture is verified for all torus and two-bridge knots.