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The study investigates how the Jones polynomial at roots of unity relates to quantum group representations, modular forms, and TQFTs.

July 29, 2026Open Access

Jones Polynomial at Roots of Unity Links Quantum Groups, Modular Forms, and TQFTs — E8 Intelligence Research

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The study investigates how the Jones polynomial at roots of unity relates to quantum group representations, modular forms, and TQFTs.
Analyzed the Jones polynomial specialized at roots of unity, particularly at q = e^{2πi/r}.
Examined representations of quantum groups, specifically U_q(sl_2), at these roots of unity.
Explored the relationship between modular forms and the action of SL(2,ℤ) on conformal blocks.
The Jones polynomial encodes quantum group representations, establishing a link between knot invariants and modular forms.
Cyclic symmetries of order r emerge from root-of-unity phases, influencing modular forms' characteristics.
Connections between graph algebras at odd-order roots and surface mapping class groups were identified.

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Finding links the Jones polynomial to quantum groups and modular forms, suggesting new connections in mathematics and physics.

Andrew Stewart Caldin (2026) studied this question.

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