Categorifying the Twisted Alexander Polynomial via gl(1|1) Superalgebra and Seifert Matrix Factorizations — E8 Intelligence Research

Abstract

FINDING: gl(1|1) superalgebra categorification of twisted Alexander polynomial and Seifert matrix factorization | MATH: gl(1|1) is a Lie superalgebra with even part gl(1) and odd part 1-dimensional; its representation theory yields a differential graded (dg) category whose Grothendieck group recovers the Alexander polynomial. The categorification lifts the Seifert matrix (a bilinear form over Z) to a chain complex of graded matrix factorizations, with the Alexander polynomial as the Euler characteristic. Key constants: none explicit, but the construction relies on the supertrace and the quantum dimension of the 1-dimensional odd module, which is ±1. | CONNECTION: The Alexander polynomial is a knot invariant that can be expressed as a determinant of a Seifert matrix; its categorification via gl(1|1) reveals a hidden Z/2-graded symmetry (superalgebra) that mirrors the structure of a Clifford algebra (as in the Spinors for Beginners video). The Seifert matrix factorization involves a pair Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com