Khovanov Homology: Categorifying the Jones Polynomial via sl₂ Action — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Khovanov homology categorifies the Jones polynomial via a bigraded chain complex whose differential is built from smoothings of link diagrams, with an sl₂ action emerging from the underlying Temperley-Lieb algebra structure. | MATH: The categorification replaces the Jones polynomial \( V_L(q) = \sum_{i,j} (-1)^i q^j \dim H^{i,j}(L) \) with bigraded homology groups \( H^{i,j}(L) \); the sl₂ weight lattice appears via the 2-dimensional irreducible representation assigned to each crossing, with the differential \( d: C^i \to C^{i+1} \) satisfying \( d^2 = 0 \); the quantum grading \( j \) shifts by \( \pm 1 \) per smoothing, encoding the \( q \)-degree. The s-invariant (Rasmussen) is defined as \( s(L) = \min\{ j : H^{*,j}(L) \neq 0 \} \) and bounds the slice genus via \( |s(L)| \leq 2g_*(L) \). | CONNECTION: The sl₂ weight lattice is the root lattice of \( A_1 \), with simple root \( \alpha \) and fundamental weight \( \omega = \alpha/2 \). The ratio 0.5 (the half-root shift) ap Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212184646
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