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

Abstract

**FINDING:** Categorification of the multi-variable Alexander polynomial using gl(1|1) superalgebra, with connections to Seifert matrix factorization and twisted Alexander invariants for abelian covers. **MATH:** - gl(1|1) superalgebra: Lie superalgebra with bosonic generator \( e \) and fermionic generators \( f, f^\dagger \) satisfying \(\{f, f^\dagger\} = e\), \([e, f] = 0\), \([e, f^\dagger] = 0\). - Alexander polynomial \(\Delta_K(t)\) for a knot \(K\): \(\Delta_K(t) = \det(t^{1/2} V - t^{-1/2} V^T)\), where \(V\) is the Seifert matrix. - Twisted Alexander polynomial: \(\Delta_K^\rho(t) = \det( t^{1/2} \rho(A) - t^{-1/2} \rho(B) )\) for a representation \(\rho: \pi_1(S^3 \setminus K) \to GL(n, \mathbb{C})\). - Categorification: Homological invariant \(\mathcal{H}_{gl(1|1)}(K)\) such that Euler characteristic \(\chi(\mathcal{H}_{gl(1|1)}(K)) = \Delta_K(t)\). - Seifert matrix factorization: \(V - V^T = \text{Seifert form}\), with \(V\) encoding linking numbers of a Seifer Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com