Linear Tridles and the Alexander Polynomial

Abstract

Yang’s linear tridle construction assigns a two-variable module to an oriented knot diagram. We show that the entire module, not only its maximal-minor polynomial, is Alexander-theoretic. A signed change of regional generators using the Alexander numbering transforms the linear tridle matrix into the Alexander–Dehn regional matrix over [Formula: see text], where [Formula: see text]. The unreduced regional module is isomorphic to [Formula: see text], where [Formula: see text] is the classical Alexander module. Consequently, the linear tridle module is its scalar extension to [Formula: see text], and all its Fitting ideals are extended Alexander ideals with an index shift of two. In particular, the maximal-minor ideal is generated by [Formula: see text]. An explicit Seifert-matrix example shows that smaller-minor ideals can distinguish knots with the same Alexander polynomial, while carrying no information beyond the Alexander module.