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Khovanov Homology and Gauge-Theoretic Extensions of the Jones Polynomial — E8 Intelligence Research

  • Andrew Stewart Caldin1
  1. 1Independent Researcher, United Kingdom
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Abstract

FINDING: Categorification of the Jones polynomial via Khovanov homology yields a bigraded chain complex whose graded Euler characteristic recovers the polynomial, with gauge-theoretic and immersed-surface extensions revealing deeper topological invariants. | MATH: Jones polynomial VL(q) satisfies skein relation q−1VL+ − qVL- = (q1/2 − q−1/2)VL0. Khovanov homology: bigraded abelian groups Hi,j(L) with χq(H*,*(L)) = VL(q) (graded Euler characteristic). Categorification replaces polynomial coefficients (integers) with homology groups (vector spaces), lifting multiplication to tensor products and direct sums. Witten's gauge-theoretic approach: Khovanov homology arises from Floer homology of a 3-manifold with link complement, using SU(2) or U(1) gauge fields; immersed surface cobordisms in ℝ4 with double points induce functorial maps between homology groups (arXiv:2510.14760v2). | CONNECTION: The graded Euler characteris

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite

BibTeXRIS
BibTeX
@misc{caldin2026khovanov,
  title = {{Khovanov Homology and Gauge-Theoretic Extensions of the Jones Polynomial — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23447069},
  url = {https://zenodo.org/records/23447069},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Khovanov Homology and Gauge-Theoretic Extensions of the Jones Polynomial — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/18
PB  - Zenodo
DO  - 10.5281/zenodo.23447069
UR  - https://zenodo.org/records/23447069
AB  - FINDING: Categorification of the Jones polynomial via Khovanov homology yields a bigraded chain complex whose graded Euler characteristic recovers the polynomial, with gauge-theoretic and immersed-surface extensions revealing deeper topological invariants. | MATH: Jones polynomial V_L(q) satisfies skein relation q^−1V_L_+ − qV_L_- = (q^1/2 − q^−1/2)V_L_0. Khovanov homology: bigraded abelian groups H^i,j(L) with χ_q(H^*,*(L)) = V_L(q) (graded Euler characteristic). Categorification replaces polynomial coefficients (integers) with homology groups (vector spaces), lifting multiplication to tensor products and direct sums. Witten's gauge-theoretic approach: Khovanov homology arises from Floer homology of a 3-manifold with link complement, using SU(2) or U(1) gauge fields; immersed surface cobordisms in ℝ^4 with double points induce functorial maps between homology groups (arXiv:2510.14760v2). | CONNECTION: The graded Euler characteris  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -